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1lgoxv6q0 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let for a triangle $$\mathrm{ABC}$$,</p>
<p>$$\overrightarrow{\mathrm{AB}}=-2 \hat{i}+\hat{j}+3 \hat{k}$$</p>
<p>$$\overrightarrow{\mathrm{CB}}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$$</p>
<p>$$\overrightarrow{\mathrm{CA}}=4 \hat{i}+3 \hat{j}+\delta \hat{k}$$</p>
<p>If $$\delta > 0$$ and the area of the tria... | [{"identifier": "A", "content": "60"}, {"identifier": "B", "content": "54"}, {"identifier": "C", "content": "120"}, {"identifier": "D", "content": "108"}] | ["A"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lh1phdby/e74db975-74e1-4987-80c4-b8ae4380dc69/bb54dde0-e665-11ed-9cfb-9bcc868d407f/file-1lh1phdbz.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lh1phdby/e74db975-74e1-4987-80c4-b8ae4380dc69/bb54dde0-e665-11ed-9cfb-9bcc868d407f/fi... | mcq | jee-main-2023-online-13th-april-evening-shift | 8,792 |
1lgq10ehr | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=3 \hat{i}+\hat{j}-\hat{k}$$ and $$\vec{c}=2 \hat{i}-3 \hat{j}+3 \hat{k}$$. If $$\vec{b}$$ is a vector such that $$\vec{a}=\vec{b} \times \vec{c}$$ and $$|\vec{b}|^{2}=50$$, then $$|72-| \vec{b}+\left.\vec{c}\right|^{2} \mid$$ is equal to __________.</p> | [] | null | 66 | <p>Given that $$\vec{a} = \vec{b} \times \vec{c}$$, we can find the magnitudes of $$\vec{a}$$ and $$\vec{c}$$:</p>
<p>$$|\vec{a}| = \sqrt{3^2 + 1^2 + (-1)^2} = \sqrt{11}$$
<br/><br/>$$|\vec{c}| = \sqrt{2^2 + (-3)^2 + 3^2} = \sqrt{22}$$</p>
<p>We know that the magnitude of the cross product of two vectors is equal to th... | integer | jee-main-2023-online-13th-april-morning-shift | 8,794 |
1lgsw25ll | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$$ and $$\vec{b}=\hat{i}+\hat{j}-\hat{k}$$. If $$\vec{c}$$ is a vector such that $$\vec{a} \cdot \vec{c}=11,
\vec{b} \cdot(\vec{a} \times \vec{c})=27$$ and $$\vec{b} \cdot \vec{c}=-\sqrt{3}|\vec{b}|$$, then $$|\vec{a} \times \vec{c}|^{2}$$ is equal to _________.</p> | [] | null | 285 | Given,
<br/><br/>$$
\begin{aligned}
& \vec{a}=\hat{i}+2 \hat{j}+3 \hat{k} \\\\
& \vec{b}=\hat{i}+\hat{j}-\hat{k} \\\\
& \vec{a} \cdot \vec{c}=11 \\\\
& \vec{b} \cdot(\vec{a} \times \vec{c})=27 \\\\
& \vec{b} \cdot \vec{c}=-\sqrt{3}|\vec{b}| \\\\
& (\vec{b} \times \vec{a}) \cdot \vec{c}=27
\end{aligned}
$$
<br/><br/>$$... | integer | jee-main-2023-online-11th-april-evening-shift | 8,795 |
1lguvx5sl | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}$$ be a non-zero vector parallel to the line of intersection of the two planes described by $$\hat{i}+\hat{j}, \hat{i}+\hat{k}$$ and $$\hat{i}-\hat{j}, \hat{j}-\hat{k}$$. If $$\theta$$ is the angle between the vector $$\vec{a}$$ and the vector $$\vec{b}=2 \hat{i}-2 \hat{j}+\hat{k}$$ and $$\vec{a} \cdot ... | [{"identifier": "A", "content": "$$\\left(\\frac{\\pi}{3}, 3 \\sqrt{6}\\right)$$"}, {"identifier": "B", "content": "$$\\left(\\frac{\\pi}{3}, 6\\right)$$"}, {"identifier": "C", "content": "$$\\left(\\frac{\\pi}{4}, 3 \\sqrt{6}\\right)$$"}, {"identifier": "D", "content": "$$\\left(\\frac{\\pi}{4}, 6\\right)$$"}] | ["D"] | null | We have, $$\vec{a}$$ is non-zero vector parallel to the line of intersection of the two planes described by $\hat{\mathbf{i}}+\hat{\mathbf{j}}, \hat{\mathbf{i}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{j}}-\hat{\mathbf{k}}$.
<br/><br/>Let $\mathbf{n}_1$ and $\mathbf{n}_2$ are the normal ve... | mcq | jee-main-2023-online-11th-april-morning-shift | 8,796 |
1lgvq0gag | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}$$ and $$\vec{c}=\hat{i}-\hat{j}+2 \hat{k}$$. Let $$\vec{d}$$ be a vector which is perpendicular to both $$\vec{a}$$ and $$\vec{b}$$, and $$\vec{c} \cdot \vec{d}=12$$. Then $$(-\hat{i}+\hat{j}-\hat{k}) \cdot(\vec{c} \times \vec{d})$$ is equal to :... | [{"identifier": "A", "content": "24"}, {"identifier": "B", "content": "42"}, {"identifier": "C", "content": "44"}, {"identifier": "D", "content": "48"}] | ["C"] | null | If $\vec{d}$ is $\perp$ to both $\vec{a}$ and $\vec{b}$ then
<br/><br/>$$
\vec{d}=\lambda(\vec{a} \times \vec{b})=\lambda\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
2 & 7 & -1 \\
3 & 0 & 5
\end{array}\right|=(35 \hat{i}-13 \hat{j}-21 \hat{k}) \lambda
$$
<br/><br/>$$
\begin{aligned}
& \text { but } \vec{c} \... | mcq | jee-main-2023-online-10th-april-evening-shift | 8,797 |
1lgxh78lc | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let O be the origin and the position vector of the point P be $$ - \widehat i - 2\widehat j + 3\widehat k$$. If the position vectors of the points A, B and C are $$ - 2\widehat i + \widehat j - 3\widehat k,2\widehat i + 4\widehat j - 2\widehat k$$ and $$ - 4\widehat i + 2\widehat j - \widehat k$$ respectively, then ... | [{"identifier": "A", "content": "$$\\frac{7}{3}$$"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "$$\\frac{10}{3}$$"}, {"identifier": "D", "content": "$$\\frac{8}{3}$$"}] | ["B"] | null | Given, the position vector of point P is :
$ \overrightarrow{OP} = -\widehat{i} - 2\widehat{j} + 3\widehat{k} $
<br/><br/>Position vectors of points A, B, and C are :
<br/><br/>$ \overrightarrow{OA} = -2\widehat{i} + \widehat{j} - 3\widehat{k} $
<br/><br/>$ \overrightarrow{OB} = 2\widehat{i} + 4\widehat{j} - 2\widehat... | mcq | jee-main-2023-online-10th-april-morning-shift | 8,798 |
1lgylle5f | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>The area of the quadrilateral $$\mathrm{ABCD}$$ with vertices $$\mathrm{A}(2,1,1), \mathrm{B}(1,2,5), \mathrm{C}(-2,-3,5)$$ and $$\mathrm{D}(1,-6,-7)$$ is equal to :</p> | [{"identifier": "A", "content": "48"}, {"identifier": "B", "content": "$$8 \\sqrt{38}$$"}, {"identifier": "C", "content": "54"}, {"identifier": "D", "content": "$$9 \\sqrt{38}$$"}] | ["B"] | null | $$
\begin{aligned}
& \text { Here } \overrightarrow{\mathrm{AC}}=(-2-2) \hat{i}+(-3-1) \hat{j}+(5-1) \hat{k} \\\\
& =-4 \hat{i}-4 \hat{j}+4 \hat{k} \\\\
& \overrightarrow{\mathrm{BD}}=(1-1) \hat{i}+(-6-2) \hat{j}+(-7-5) \hat{k} \\\\
& =-8 \hat{j}-12 \hat{k}
\end{aligned}
$$
<br/><br/>So, area of quadrilateral $=\frac{1... | mcq | jee-main-2023-online-8th-april-evening-shift | 8,799 |
1lh00gxyv | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}$$ and $$\vec{c}$$ be vectors such that $$\vec{a} \times \vec{c}=\vec{a} \times \vec{b}$$. If <br/><br/>$$\vec{a} \cdot \vec{c}=-12, \vec{c} \cdot(\hat{i}-2 \hat{j}+\hat{k})=5$$, then $$\vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})$$ i... | [] | null | 11 | Let $\vec{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$
<br/><br/>Now, $\vec{a} \cdot \vec{c}=-12$
<br/><br/>$$
\Rightarrow 6 c_1+9 c_2+12 c_3=-12
$$ ..............(i)
<br/><br/>Also, $\vec{c} \cdot(\hat{i}-2 \hat{j}+\hat{k})=5$
<br/><br/>$$
\Rightarrow c_1-2 c_2+c_3=5
$$ ................(ii)
<br/><br/>$$
\begin{aligned}
& \... | integer | jee-main-2023-online-8th-april-morning-shift | 8,800 |
lsaojnb9 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $\overrightarrow{\mathrm{a}}=-5 \hat{i}+\hat{j}-3 \hat{k}, \overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}-4 \hat{k}$ and
<br/><br/>$\overrightarrow{\mathrm{c}}=(((\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \hat{i}) \times \hat{i}) \times \hat{i}$. Then $\vec{c} \cdot(-\hat{i}+\hat{j}+\ha... | [{"identifier": "A", "content": "-12"}, {"identifier": "B", "content": "-10"}, {"identifier": "C", "content": "-13"}, {"identifier": "D", "content": "-15"}] | ["A"] | null | $\begin{aligned} & \vec{a}=-5 \cdot \hat{i}+\hat{j}-3 \hat{k}, \vec{b}=\hat{i}+2 \hat{j}-4 \hat{k} \\\\ & \vec{c}=(((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i} \\\\ & =(((\vec{a} \cdot \hat{i}) \vec{b}-(\vec{b} \cdot \hat{i}) \vec{a}) \times \hat{i}) \times \hat{i} \\\\ & =((-5 \vec{b}-\vec{... | mcq | jee-main-2024-online-1st-february-morning-shift | 8,803 |
jaoe38c1lscmwyjl | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let the position vectors of the vertices $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}$$ of a triangle be $$2 \hat{i}+2 \hat{j}+\hat{k}, \hat{i}+2 \hat{j}+2 \hat{k}$$ and $$2 \hat{i}+\hat{j}+2 \hat{k}$$ respectively. Let $$l_1, l_2$$ and $$l_3$$ be the lengths of perpendiculars drawn from the ortho center of the trian... | [{"identifier": "A", "content": "$$\\frac{1}{4}$$"}, {"identifier": "B", "content": "$$\\frac{1}{5}$$"}, {"identifier": "C", "content": "$$\\frac{1}{3}$$"}, {"identifier": "D", "content": "$$\\frac{1}{2}$$"}] | ["D"] | null | <p>$$\triangle \mathrm{ABC}$$ is equilateral</p>
<p>Orthocentre and centroid will be same</p>
<p>$$\mathrm{G}\left(\frac{5}{3}, \frac{5}{3}, \frac{5}{3}\right)$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lt1vv91f/efa66592-d248-4f32-8e31-2416eaa514c4/4caa5d30-d411-11ee-b9d5-0585032231f0/fil... | mcq | jee-main-2024-online-27th-january-evening-shift | 8,804 |
jaoe38c1lsd4wwd2 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=3 \hat{i}+2 \hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$$ and $$\vec{c}$$ be a vector such that $$(\vec{a}+\vec{b}) \times \vec{c}=2(\vec{a} \times \vec{b})+24 \hat{j}-6 \hat{k}$$ and $$(\vec{a}-\vec{b}+\hat{i}) \cdot \vec{c}=-3$$. Then $$|\vec{c}|^2$$ is equal to ________.</p> | [] | null | 38 | <p>$$\begin{aligned}
& (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}=2(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})+24 \hat{\mathrm{j}}-6 \hat{\mathrm{k}} \\
& (5 \hat{\mathrm{i}}+\hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \times \overrightarrow{\mathrm{c}}=2(... | integer | jee-main-2024-online-31st-january-evening-shift | 8,805 |
jaoe38c1lse565zc | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=4 \hat{i}+\hat{j}+7 \hat{k}$$ and $$\vec{c}=\hat{i}-3 \hat{j}+4 \hat{k}$$ be three vectors. If a vectors $$\vec{p}$$ satisfies $$\vec{p} \times \vec{b}=\vec{c} \times \vec{b}$$ and $$\vec{p} \cdot \vec{a}=0$$, then $$\vec{p} \cdot(\hat{i}-\hat{j}-\hat{k})$$ is equal... | [{"identifier": "A", "content": "24"}, {"identifier": "B", "content": "32"}, {"identifier": "C", "content": "36"}, {"identifier": "D", "content": "28"}] | ["B"] | null | <p>$$\begin{aligned}
& \overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{0} \\
& (\overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}=\overrightarrow{0} \\
& \overrightarrow{\mathrm{p}}-\o... | mcq | jee-main-2024-online-31st-january-morning-shift | 8,806 |
jaoe38c1lse59x61 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>The distance of the point $$Q(0,2,-2)$$ form the line passing through the point $$P(5,-4, 3)$$ and perpendicular to the lines $$\vec{r}=(-3 \hat{i}+2 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+5 \hat{k}), \lambda \in \mathbb{R}$$ and $$\vec{r}=(\hat{i}-2 \hat{j}+\hat{k})+\mu(-\hat{i}+3 \hat{j}+2 \hat{k}), \mu \in \mathbb{... | [{"identifier": "A", "content": "$$\\sqrt{74}$$\n"}, {"identifier": "B", "content": "$$\\sqrt{86}$$\n"}, {"identifier": "C", "content": "$$\\sqrt{54}$$\n"}, {"identifier": "D", "content": "$$\\sqrt{20}$$"}] | ["A"] | null | <p>A vector in the direction of the required line can be obtained by cross product of</p>
<p>$$\begin{aligned}
& \left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
2 & 3 & 5 \\
-1 & 3 & 2
\end{array}\right| \\\\
& =-9 \hat{i}-9 \hat{j}+9 \hat{k}
\end{aligned}$$</p>
<p>Required line<... | mcq | jee-main-2024-online-31st-january-morning-shift | 8,807 |
jaoe38c1lse5sysr | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}|=1,|\vec{b}|=4$$, and $$\vec{a} \cdot \vec{b}=2$$. If $$\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$$ and the angle between $$\vec{b}$$ and $$\vec{c}$$ is $$\alpha$$, then $$192 \sin ^2 \alpha$$ is equal to ________.</p> | [] | null | 48 | <p>$$\begin{aligned}
& \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=(2 \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \cdot \overrightarrow{\mathrm{b}}-3|\mathrm{b}|^2 \\
& |\mathrm{~b}||c| \cos \alpha=-3|\mathrm{~b}|^2 \\
& |\mathrm{c}| \cos \alpha=-12 \text {, as }|\mathrm{b}|=4 \\
&... | integer | jee-main-2024-online-31st-january-morning-shift | 8,808 |
jaoe38c1lsfkro4t | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=12 \vec{a}+4 \vec{b} \text { and } \overrightarrow{O C}=\vec{b}$$, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then $$\mathrm{{{area\,of\,the\,quadrilateral\,OA\,BC} \over {area\,of\,S}}}$$ is equal to _________.</p> | [{"identifier": "A", "content": "7"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "8"}, {"identifier": "D", "content": "10"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsr8xnpy/836457b4-b43d-438c-b5ce-783fa67cdc3d/c728ff60-ce37-11ee-9412-cd4f9c6f2c40/file-6y3zli1lsr8xnpz.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsr8xnpy/836457b4-b43d-438c-b5ce-783fa67cdc3d/c728ff60-ce37-11ee... | mcq | jee-main-2024-online-29th-january-evening-shift | 8,809 |
1lsg4dytc | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in \mathbb{R}$$. Let a vector $$\vec{b}$$ be such that the angle between $$\vec{a}$$ and $$\vec{b}$$ is $$\frac{\pi}{4}$$ and $$|\vec{b}|^2=6$$. If $$\vec{a} \cdot \vec{b}=3 \sqrt{2}$$, then the value of $$\left(\alpha^2+\beta^2\right)|\vec{a} \times... | [{"identifier": "A", "content": "85"}, {"identifier": "B", "content": "90"}, {"identifier": "C", "content": "75"}, {"identifier": "D", "content": "95"}] | ["B"] | null | <p>$$\begin{aligned}
& |\overrightarrow{\mathrm{b}}|^2=6 ;|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \cos \theta=3 \sqrt{2} \\
& |\overrightarrow{\mathrm{a}}|^2|\overrightarrow{\mathrm{b}}|^2 \cos ^2 \theta=18 \\
& |\overrightarrow{\mathrm{a}}|^2=6
\end{aligned}$$</p>
<p>Also $$1+\alpha^2+\beta^2=6$$</p... | mcq | jee-main-2024-online-30th-january-evening-shift | 8,810 |
luxwcrzd | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Between the following two statements:</p>
<p>Statement I : Let $$\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$$ and $$\vec{b}=2 \hat{i}+\hat{j}-\hat{k}$$. Then the vector $$\vec{r}$$ satisfying $$\vec{a} \times \vec{r}=\vec{a} \times \vec{b}$$ and $$\vec{a} \cdot \vec{r}=0$$ is of magnitude $$\sqrt{10}$$.</p>
<p>Statement II... | [{"identifier": "A", "content": "Both Statement I and Statement II are correct.\n"}, {"identifier": "B", "content": "Both Statement I and Statement II are incorrect.\n"}, {"identifier": "C", "content": "Statement I is correct but Statement II is incorrect.\n"}, {"identifier": "D", "content": "Statement I is incorrect b... | ["D"] | null | <p>$$\begin{aligned}
& \because \quad \forall \text { two vectors } \vec{c} \text { & } \vec{d} \\
& |\vec{c} \times \vec{d}|^2=|\vec{c}|^2|\vec{d}|^2-(\vec{c} \cdot \vec{d})^2 \\
& \text { replacing } \vec{c}=\vec{a} ~\& ~\vec{d}=\vec{r} \\
& \Rightarrow|\vec{a} \times \vec{r}|=|\vec{a}|^2|\vec{r}|^2-(\vec{a} \cdot \v... | mcq | jee-main-2024-online-9th-april-evening-shift | 8,813 |
luxwe3dl | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=2 \hat{i}+\alpha \hat{j}+\hat{k}, \vec{b}=-\hat{i}+\hat{k}, \vec{c}=\beta \hat{j}-\hat{k}$$, where $$\alpha$$ and $$\beta$$ are integers and $$\alpha \beta=-6$$. Let the values of the ordered pair $$(\alpha, \beta)$$, for which the area of the parallelogram of diagonals $$\vec{a}+\vec{b}$$ and $$\vec{b... | [{"identifier": "A", "content": "21"}, {"identifier": "B", "content": "24"}, {"identifier": "C", "content": "19"}, {"identifier": "D", "content": "17"}] | ["C"] | null | <p>Area of parallelogram whose diagonals are $$\vec{a}+\vec{b}$$ and $$\vec{b}+\vec{c}$$ is</p>
<p>$$\begin{aligned}
& =\frac{1}{2}|(\vec{a}+\vec{b}) \times(\vec{b}+\vec{c})| \\
& =\frac{1}{2}|\vec{a} \times \vec{b}+\vec{a} \times \vec{c}+\vec{b} \times \vec{c}| \\
& =\frac{1}{2}|-2 \beta \hat{i}-2 \hat{j}+(\alpha+\bet... | mcq | jee-main-2024-online-9th-april-evening-shift | 8,814 |
luy6z4lq | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let three vectors ,$$\overrightarrow{\mathrm{a}}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{b}}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \overrightarrow{\mathrm{c}}=x \hat{i}+y \hat{j}+z \hat{k}$$ form a triangle such that $$\vec{c}=\vec{a}-\vec{b}$$ and the area of the triangle is $$5 \sqrt{6}$$. If $$\alpha... | [{"identifier": "A", "content": "14"}, {"identifier": "B", "content": "12"}, {"identifier": "C", "content": "16"}, {"identifier": "D", "content": "10"}] | ["A"] | null | <p>To solve this, let's start with the given vector equation:</p>
<p>$$\vec{c} = \vec{a} - \vec{b}$$</p>
<p>Given vectors are:</p>
<p>$$\overrightarrow{\mathrm{a}} = \alpha \hat{i} + 4 \hat{j} + 2 \hat{k}$$</p>
<p>$$\overrightarrow{\mathrm{b}} = 5 \hat{i} + 3 \hat{j} + 4 \hat{k}$$</p>
<p>Then, the vector $$\overri... | mcq | jee-main-2024-online-9th-april-morning-shift | 8,815 |
luy6z5a5 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow{O A}=2 \vec{a}, \overrightarrow{O B}=6 \vec{a}+5 \vec{b}$$ and $$\overrightarrow{O C}=3 \vec{b}$$, where $$O$$ is the origin. If the area of the parallelogram with adjacent sides $$\overrightarrow{O A}$$ and $$\overrightarrow{O C}$$ is 15 sq. units, then the area (in sq. units) of the quadrilat... | [{"identifier": "A", "content": "32"}, {"identifier": "B", "content": "38"}, {"identifier": "C", "content": "35"}, {"identifier": "D", "content": "40"}] | ["C"] | null | <p>$$\begin{aligned}
& 6|\vec{a} \times \vec{b}|=15 \\
& \Rightarrow|\vec{a} \times \vec{b}|=\frac{5}{2}
\end{aligned}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/jaoe38c1lw3cizlj/32ca14d9-c7d9-4752-a76d-bc023959c231/e1a4ec70-1043-11ef-9f6c-75804a813f04/file-jaoe38c1lw3cizlk.png?for... | mcq | jee-main-2024-online-9th-april-morning-shift | 8,816 |
lv0vxdun | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\mathrm{ABC}$$ be a triangle of area $$15 \sqrt{2}$$ and the vectors $$\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \overrightarrow{\mathrm{BC}}=\mathrm{a} \hat{i}+\mathrm{b} \hat{j}+\mathrm{c} \hat{k}$$ and $$\overrightarrow{\mathrm{AC}}=6 \hat{i}+\mathrm{d} \hat{j}-2 \hat{k}, \mathrm{~d}>0$$.... | [] | null | 54 | <p>Area of triangle $$A B C=15 \sqrt{2}$$</p>
<p>$$\begin{aligned}
& \Rightarrow \frac{1}{2}|\overline{A B} \times \overline{A C}|=15 \sqrt{2} \quad \text{.... (i)}\\
& \quad \overline{A B} \times \overline{A C}\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
1 & 2 & -7 \\
6 & d & -2
\end{array}\right| \\
& =(7 ... | integer | jee-main-2024-online-4th-april-morning-shift | 8,817 |
lv7v4g3o | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>If $$\mathrm{A}(1,-1,2), \mathrm{B}(5,7,-6), \mathrm{C}(3,4,-10)$$ and $$\mathrm{D}(-1,-4,-2)$$ are the vertices of a quadrilateral ABCD, then its area is :</p> | [{"identifier": "A", "content": "$$24 \\sqrt{7}$$\n"}, {"identifier": "B", "content": "$$48 \\sqrt{7}$$\n"}, {"identifier": "C", "content": "$$24 \\sqrt{29}$$\n"}, {"identifier": "D", "content": "$$12 \\sqrt{29}$$"}] | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwgfpgex/f708b26f-9a53-4063-a888-8e61f5ee5884/8151b490-1776-11ef-bded-616e4abb66b9/file-1lwgfpgey.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwgfpgex/f708b26f-9a53-4063-a888-8e61f5ee5884/8151b490-1776-11ef-bded-616e4abb66b9... | mcq | jee-main-2024-online-5th-april-morning-shift | 8,820 |
lv7v3oem | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow{\mathrm{a}}=\hat{i}-3 \hat{j}+7 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}-\hat{j}+\hat{k}$$ and $$\overrightarrow{\mathrm{c}}$$ be a vector such that $$(\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}=3(\overrightarrow{\mathrm{c}} \times \overr... | [] | null | 30 | <p>$$(\vec{a}+2 \vec{b}) \times \vec{c}=3(\vec{c} \times \vec{a})$$</p>
<p>$$\begin{aligned}
\Rightarrow \quad & \vec{b} \times \vec{c}+2(\vec{a} \times \vec{c})=0 \\
& (\vec{b}+2 \vec{a}) \times \vec{c}=0 \\
& \vec{c}=\lambda(\vec{b}+2 \vec{a}) \\
& \vec{c} \cdot \vec{a}=130 \Rightarrow \lambda=1 \\
& \vec{c}=4 \hat{i... | integer | jee-main-2024-online-5th-april-morning-shift | 8,821 |
lv9s2007 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=2 \hat{i}+5 \hat{j}-\hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}+2 \hat{k}$$ and $$\vec{c}$$ be three vectors such that $$(\vec{c}+\hat{i}) \times(\vec{a}+\vec{b}+\hat{i})=\vec{a} \times(\vec{c}+\hat{i})$$. If $$\vec{a} \cdot \vec{c}=-29$$, then $$\vec{c} \cdot(-2 \hat{i}+\hat{j}+\hat{k})$$ is equal to:</p> | [{"identifier": "A", "content": "15"}, {"identifier": "B", "content": "10"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "12"}] | ["C"] | null | <p>$$\begin{gathered}
(\vec{c}+\hat{i}) \times(\vec{a}+\vec{b}+\hat{i}+\vec{a})=0 \\
\Rightarrow \quad \vec{c}+\hat{i}=\lambda(\vec{a}+\vec{b}+\hat{i}+\vec{a}) \\
=\lambda(2 \vec{a}+\vec{b}+\hat{i}) \\
\quad=\lambda(7 \hat{i}+8 \hat{j}) \\
\Rightarrow \quad \vec{c}=(7 \lambda-1) \hat{i}+8 \lambda \hat{j} \\
\quad \vec{... | mcq | jee-main-2024-online-5th-april-evening-shift | 8,822 |
lvb294j7 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow{\mathrm{a}}=6 \hat{i}+\hat{j}-\hat{k}$$ and $$\overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}$$. If $$\overrightarrow{\mathrm{c}}$$ is a is vector such that $$|\overrightarrow{\mathrm{c}}| \geq 6, \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=6|\overrightarrow{\mathrm{c}}|,|\overrig... | [{"identifier": "A", "content": "$$\\frac{3}{2} \\sqrt{6}$$\n"}, {"identifier": "B", "content": "$$\\frac{9}{2}(6-\\sqrt{6})$$\n"}, {"identifier": "C", "content": "$$\\frac{9}{2}(6+\\sqrt{6})$$\n"}, {"identifier": "D", "content": "$$\\frac{3}{2} \\sqrt{3}$$"}] | ["C"] | null | <p>$$\begin{aligned}
& |(\vec{a} \times \vec{b}) \times \vec{c}|=|\vec{a} \times \vec{b}||\vec{c}| \sin 60^{\circ} \\
& \left|\begin{array}{ccc}
i & j & k \\
6 & 1 & -1 \\
1 & 1 & 0
\end{array}\right|=i(1)-j(1)+k(5) \\
& =i-j+5 k \\
& |\vec{a} \times \vec{b}|=\sqrt{1+1+25}=\sqrt{27} \\
& |\vec{c}-\vec{a}|=2 \sqrt{2} \\... | mcq | jee-main-2024-online-6th-april-evening-shift | 8,825 |
nPwDAPpBUUrYKsRB | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | The power factor of $$AC$$ circuit having resistance $$(R)$$ and inductance $$(L)$$ connected in series and an angular velocity $$\omega $$ is | [{"identifier": "A", "content": "$$R/\\omega L$$ "}, {"identifier": "B", "content": "$$R/{\\left( {{R^2} + {\\omega ^2}{L^2}} \\right)^{1/2}}$$ "}, {"identifier": "C", "content": "$$\\omega L/R$$ "}, {"identifier": "D", "content": "$$R/{\\left( {{R^2} - {\\omega ^2}{L^2}} \\right)^{1/2}}$$ "}] | ["B"] | null | The impedance triangle for resistance $$\left( R \right)$$ and inductor $$(L)$$ connected in series is shown in the figure.
<br><br><img class="question-image" src="https://imagex.cdn.examgoal.net/EyI9jwNfZakNSnx6N/RG2tUhYR6L2pFM5r986jI6AkPlnGj/HFxkeghaZOPffeRH5uOmx1/image.svg" loading="lazy" alt="AIEEE 2002 Physics - ... | mcq | aieee-2002 | 8,827 |
lydY8ehUtKC26aZ0 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In an $$LCR$$ series $$a.c.$$ circuit, the voltage across each of the components, $$L,C$$ and $$R$$ is $$50V$$. The voltage across the $$L.C$$ combination will be : | [{"identifier": "A", "content": "$$100V$$ "}, {"identifier": "B", "content": "$$50\\sqrt 2 $$ "}, {"identifier": "C", "content": "$$50$$ $$V$$ "}, {"identifier": "D", "content": "$$0$$ $$V$$ (zero) "}] | ["D"] | null | Since the phase difference between $$L$$ & $$C$$ is $$\pi ,$$
<br><br>$$\therefore$$ net voltage difference across $$LC=50-50=0$$ | mcq | aieee-2004 | 8,828 |
oi26w9o1iXOCTXgv | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | Alternating current can not be measured by $$D.C.$$ ammeter because | [{"identifier": "A", "content": "Average value of current for complete cycle is zero "}, {"identifier": "B", "content": "$$A.C.$$ Changes direction "}, {"identifier": "C", "content": "$$A.C.$$ can not pass through $$D.C.$$ Ammeter "}, {"identifier": "D", "content": "$$D.C.$$ Ammeter will get damaged. "}] | ["A"] | null | $$D.C.$$ ammeter measure average current in $$AC$$ current, average current is zero for complete cycle. Hence reading will be zero. | mcq | aieee-2004 | 8,830 |
YEww4HpdQL4O8otA | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | The self inductance of the motor of an electric fan is $$10$$ $$H$$. In order to impart maximum power at $$50$$ $$Hz$$, it should be connected to a capacitance of | [{"identifier": "A", "content": "$$8\\mu F$$ "}, {"identifier": "B", "content": "$$4\\mu F$$"}, {"identifier": "C", "content": "$$2\\mu F$$"}, {"identifier": "D", "content": "$$1\\mu F$$"}] | ["D"] | null | For maximum power, $${X_L} = X{}_C,$$ which yields
<br><br>$$C = {1 \over {{{\left( {2\pi n} \right)}^2}L}} = {1 \over {4{\pi ^2} \times 50 \times 50 \times 10}}$$
<br><br>$$\therefore$$ $$C = 0.1 \times {10^{ - 5}}F = 1\mu F$$ | mcq | aieee-2005 | 8,831 |
iLTAg7jdqZg8KmtF | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A circuit has a resistance of $$12$$ $$ohm$$ and an impedance of $$15$$ $$ohm$$. The power factor of the circuit will be | [{"identifier": "A", "content": "$$0.4$$ "}, {"identifier": "B", "content": "$$0.8$$ "}, {"identifier": "C", "content": "$$0.125$$ "}, {"identifier": "D", "content": "$$1.25$$ "}] | ["B"] | null | Power factor $$ = \cos \phi = {R \over Z} = {{12} \over {15}} = {4 \over 5} = 0.8$$ | mcq | aieee-2005 | 8,832 |
uRbFUgJGpog2DiZu | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | The phase difference between the alternating current and $$emf$$ is $${\pi \over 2}.$$ Which of the following cannot be the constituent of the circuit? | [{"identifier": "A", "content": "$$R,L$$ "}, {"identifier": "B", "content": "$$C$$ alone "}, {"identifier": "C", "content": "$$L$$ alone "}, {"identifier": "D", "content": "$$L, C$$ "}] | ["A"] | null | <p>The phase difference between the alternating current and emf in an AC circuit depends on the components in the circuit:</p>
<ul>
<li>In a purely resistive ($R$) circuit, the current and emf are in phase, meaning the phase difference is $0$.</li><br/>
<li>In a purely inductive ($L$) circuit, the current lags behind t... | mcq | aieee-2005 | 8,833 |
ASzIjiBCgAA6c2TN | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In a series resonant $$LCR$$ circuit, the voltage across $$R$$ is $$100$$ volts and $$R = 1\,k\Omega $$ with $$C = 2\mu F.$$ The resonant frequency $$\omega $$ is $$200$$ $$rad/s$$. At resonance the voltage across $$L$$ is | [{"identifier": "A", "content": "$$2.5 \\times {10^{ - 2}}V$$ "}, {"identifier": "B", "content": "$$40$$ $$V$$ "}, {"identifier": "C", "content": "$$250$$ $$V$$ "}, {"identifier": "D", "content": "$$4 \\times {10^{ - 3}}V$$ "}] | ["C"] | null | Across resistor, $$I = {V \over R} = {{100} \over {1000}} = 0.1A$$
<br><br>At resonance,
<br><br>$${X_L} = {X_C} = {1 \over {\omega C}}$$
<br><br>$$ = {1 \over {200 \times 2 \times {{10}^{ - 6}}}} = 2500$$
<br><br>Voltage across $$L$$ is
<br><br>$$I{X_L} = 0.1 \times 2500 = 250V$$ | mcq | aieee-2006 | 8,834 |
cW8qYk6sz4VQ7ASe | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In an $$a.c.$$ circuit the voltage applied is $$E = {E_0}\,\sin \,\omega t.$$ The resulting current in the circuit is $$I = {I_0}\sin \left( {\omega t - {\pi \over 2}} \right).$$ The power consumption in the circuit is given by | [{"identifier": "A", "content": "$$P = \\sqrt 2 {E_0}{I_0}$$ "}, {"identifier": "B", "content": "$$P = {{{E_0}{I_0}} \\over {\\sqrt 2 }}$$ "}, {"identifier": "C", "content": "$$P=zero$$ "}, {"identifier": "D", "content": "$$P = {{{E_0}{I_0}} \\over 2}$$ "}] | ["C"] | null | <b>KEY CONCEPT : </b> We know that power consumed in a.c. circuit is given by,
<br><br>$$P = {E_{rms}}{I_{rms}}\cos \phi $$
<br><br>Here, $$E = {E_0}\sin \omega t$$
<br><br>$$I = {I_0}\sin \left( {\omega t - {\pi \over 2}} \right)$$
<br><br>which implies that the phase difference, $$\phi = {\pi \over 2}$$
<br><br>$... | mcq | aieee-2007 | 8,835 |
thulUeTMEHlNKBZ2 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In a series $$LCR$$ circuit $$R = 200\Omega $$ and the voltage and the frequency of the main supply is $$220V$$ and $$50$$ $$Hz$$ respectively. On taking out the capacitance from the circuit the current lags behind the voltage by $${30^ \circ }.$$ On taking out the inductor from the circuit the current leads the voltag... | [{"identifier": "A", "content": "$$305$$ $$W$$ "}, {"identifier": "B", "content": "$$210$$ $$W$$ "}, {"identifier": "C", "content": "$$zero$$ $$W$$ "}, {"identifier": "D", "content": "$$242$$ $$W$$ "}] | ["D"] | null | When capacitance is taken out, the circular is $$LR.$$
<br><br>$$\therefore$$ $$\tan \phi = {{\omega L} \over R}$$
<br><br>$$ \Rightarrow \omega L = R\,\tan \phi $$
<br><br>$$ = 200 \times {1 \over {\sqrt 3 }} = {{200} \over {\sqrt 3 }}$$
<br><br>Again, when inductor is taken out, the circuit is $$CR.$$
<br><br>$$\th... | mcq | aieee-2010 | 8,836 |
qbVVQJ5xx7tlSAN0 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A fully charged capacitor $$C$$ with initial charge $${q_0}$$ is connected to a coil of self inductance $$L$$ at $$t=0.$$ The time at which the energy is stored equally between the electric and the magnetic fields is : | [{"identifier": "A", "content": "$${\\pi \\over 4}\\sqrt {LC} $$ "}, {"identifier": "B", "content": "$$2\\pi \\sqrt {LC} $$ "}, {"identifier": "C", "content": "$$\\sqrt {LC} $$ "}, {"identifier": "D", "content": "$$\\pi \\sqrt {LC} $$ "}] | ["A"] | null | Energy stored in magnetic field $$ = {1 \over 2}L{i^2}$$
<br><br>Energy stored in electric field $$ = {1 \over 2}{{{q^2}} \over C}$$
<br><br>$$\therefore$$ $${1 \over 2}L{i^2} = {1 \over 2}{{{q^2}} \over C}$$
<br><br>Also $$q = {q_0}\,\cos \,\omega t$$ and $$\omega = {1 \over {\sqrt {LC} }}$$
<br><br>On solving $$t = ... | mcq | aieee-2011 | 8,837 |
h17MQANUOWM7bVSn9Bq9s | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | An arc lamp requires a direct current of
10 A at 80 V to function. If it is connected
to a 220 V (rms), 50 Hz AC supply, the
series inductor needed for it to work is
close to : | [{"identifier": "A", "content": "0.044 H"}, {"identifier": "B", "content": "0.065 H"}, {"identifier": "C", "content": "80 H"}, {"identifier": "D", "content": "0.08 H"}] | ["B"] | null | <p>From the circuit, we have</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l36s2v57/7fb9ba2f-2acb-4af2-95e3-05d177745de7/c748d6c0-d404-11ec-b808-5752a3163b13/file-1l36s2v58.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l36s2v57/7fb9ba2f-2acb-4af2-95e3-05d177745de7/c74... | mcq | jee-main-2016-offline | 8,838 |
wwtanXWySkKGKkJvWaaD7 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A sinusoidal voltage of peak value 283 V and angular frequency 320/s is applied to a series LCR circuit. Given that R=5 $$\Omega $$, L=25 mH and C=1000 $$\mu $$F. The total impedance, and phase difference between the voltage across the source and the current will respectively be : | [{"identifier": "A", "content": "10 $$\\Omega $$ and tan<sup>$$-$$1</sup> $$\\left( {{5 \\over 3}} \\right)$$"}, {"identifier": "B", "content": "$$7\\,\\Omega $$ and 45<sup>o</sup> "}, {"identifier": "C", "content": "$$10\\,\\Omega $$ and tan<sup>$$-$$1</sup>$$\\left( {{8 \\over 3}} \\right)$$"}, {"identifier": "D", "... | ["B"] | null | <p>It is given that e<sub>0</sub> = 283 V; $$\omega$$ = 320.</p>
<p>The inductor reactance is X<sub>L</sub> = 320 $$\times$$ 25 $$\times$$ 10<sup>$$-$$3</sup> = 8 $$\Omega$$</p>
<p>The capacitor reactance is</p>
<p>$${X_C} = {1 \over {\omega C}} = {1 \over {320 \times 1000 \times {{10}^{ - 6}}}} = {{1000} \over {320}} ... | mcq | jee-main-2017-online-9th-april-morning-slot | 8,839 |
wfr9c3vxd7eO8KhD | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In an a.c. circuit, the instantaneous e.m.f. and current are given by <br/>
e = 100 sin 30 t<br/>
i = 20 sin $$\left( {30t - {\pi \over 4}} \right)$$<br/>
In one cycle of a.c., the average power consumed by the circuit and the wattless current are, respectively | [{"identifier": "A", "content": "50, 0 "}, {"identifier": "B", "content": "50, 10 "}, {"identifier": "C", "content": "$${{1000} \\over {\\sqrt 2 }},10$$ "}, {"identifier": "D", "content": "$${{50} \\over {\\sqrt 2 }}$$ "}] | ["C"] | null | Wattless current,
<br><br>here $$\phi $$ is the angle between i and e.
<br><br>Average power,
<br><br>P<sub>av</sub> = V<sub>rms</sub> I<sub>rms</sub> cos$$\phi $$
<br><br>= $${{100} \over {\sqrt 2 }} \times {{20} \over {\sqrt 2 }}$$ cos$${\pi \over 4}$$
<br><br>= $${{1000} \over {\sqrt 2 }}$$ wa... | mcq | jee-main-2018-offline | 8,840 |
LErmRAs7aPfk9kZnjbEJN | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | An ideal capacitor of capacitance $$0.2\,\mu F$$ is charged to a potential difference of $$10$$ $$V.$$ The charging battery is then disconnected. The capacitor is then connected to an ideal inductor of self inductance $$0.5$$ $$mH.$$ The current at a time when the potential difference across the capacitor is $$5$$ $$V,... | [{"identifier": "A", "content": "$$0.34\\,\\,A$$ "}, {"identifier": "B", "content": "$$0.25\\,\\,A$$"}, {"identifier": "C", "content": "$$0.17\\,\\,A$$"}, {"identifier": "D", "content": "$$0.15\\,\\,A$$"}] | ["C"] | null | Capacitance, C = 0.2 $$\mu $$F = 0.2 $$ \times $$ 10<sup>$$-$$6</sup> F
<br><br>Inductance, L = 0.5 m H = 0.5 $$ \times $$ 10<sup>$$-$$3</sup> H
<br><br>Let, current = I.
<br><br>Using energy conservation,
<br><br>U<sub>E</sub> + 0 = U<sub>E</sub><sup>'</sup> + U<sub>b</sub><sup>'</sup>
<br><br>$$ \Rightarrow $$$$\... | mcq | jee-main-2018-online-15th-april-morning-slot | 8,841 |
qfDj6DNEGPaAx4oS19SbE | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A circuit connected to an ac source of emf
e = e<sub>0</sub>sin(100t) with t in seconds, gives a phase
difference of $$\pi $$/4 between the emf e and
current i. Which of the following circuits will
exhibit this ? | [{"identifier": "A", "content": "RC circuit with R = 1 k$$\\Omega $$ and C = 1\u03bcF"}, {"identifier": "B", "content": "RL circuit with R = 1k$$\\Omega $$ and L = 1mH"}, {"identifier": "C", "content": "RC circuit with R = 1k$$\\Omega $$ and C = 10 \u03bcF"}, {"identifier": "D", "content": "RL circuit with R = 1 k$$\\O... | ["C"] | null | Given phase difference = $${\pi \over 4}$$ and $$\omega $$ = 100 rad/s<br><br>
$$ \Rightarrow $$ Reactance (X) = Resistance (R)
Now by checking option. <br><br>
Option (A)<br>
R = 1000 $$\Omega $$ and X<sub>c</sub> = $${1 \over {{{10}^{ - 6}} \times 100}} = {10^4}\Omega $$<br><br>
Option (B)<br>
R = 10<sup>3</sup> $$... | mcq | jee-main-2019-online-8th-april-evening-slot | 8,842 |
SxYBTBePrEFrTUvxCfLI5 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | An alternating voltage v(t) = 220 sin 100 $$\pi $$t volt
is applied to a purely resistance load of 50$$\Omega $$ .
The time taken for the current to rise from half
of the peak value to the peak value is : | [{"identifier": "A", "content": "5 ms"}, {"identifier": "B", "content": "2.2 ms"}, {"identifier": "C", "content": "3.3 ms"}, {"identifier": "D", "content": "7.2 ms"}] | ["C"] | null | <p>In an AC resistive circuit, current and voltage are in phase.</p>
<p>So, $$I = {V \over R} \Rightarrow I = {{220} \over {50}}\sin (100\pi t)$$ ..... (i)</p>
<p>$$\therefore$$ Time period of one complete cycle of current is</p>
<p>$$T = {{2\pi } \over \omega } = {{2\pi } \over {100\pi }} = {1 \over {50}}s$$</p>
<p><i... | mcq | jee-main-2019-online-8th-april-morning-slot | 8,843 |
HUVQ6DDsSGbrT6U5Fs08q | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In the above circuit, C = $${{\sqrt 3 } \over 2}$$$$\mu $$F, R<sub>2</sub> = 20 $$\Omega $$, L = $${{\sqrt 3 } \over {10}}$$ H and R<sub>1</sub> = 10 $$\Omega $$. Current in L-R<sub>1</sub> path is I<sub>1</sub> and in C-R<sub>2</sub> path it is I<sub>2</sub> . The voltage of A.C. source is given by, V = 200 $${\sqrt 2... | [{"identifier": "A", "content": "150<sup>o</sup>"}, {"identifier": "B", "content": "90<sup>o</sup>"}, {"identifier": "C", "content": "30<sup>o</sup>"}, {"identifier": "D", "content": "0<sup>o</sup>"}] | ["C"] | null | <p>Phase difference between I<sub>2</sub> and V, i.e. C $$-$$ R<sub>2</sub> circuit is given by</p>
<p>$$\tan \phi = {{{X_C}} \over {{R_2}}} \Rightarrow \tan \phi = {1 \over {C\omega {R_2}}}$$</p>
<p>Substituting the given values, we get</p>
<p>$$\tan \phi = {1 \over {{{\sqrt 3 } \over 2} \times {{10}^{ - 6}} \times... | mcq | jee-main-2019-online-12th-january-evening-slot | 8,844 |
kOvprkYFwIeXUlt3fP03J | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A series AC circuit containing an inductor (20 mH), a capacitor (120 $$\mu $$F) and a resistor (60 $$\Omega $$) is driven by an AC source of 24V/50 Hz. The energy dissipated in the circuit in 60 s is : | [{"identifier": "A", "content": "5.65 $$ \\times $$ 10<sup>2</sup>J "}, {"identifier": "B", "content": "2.26 $$ \\times $$ 10<sup>3</sup>J"}, {"identifier": "C", "content": "5.17 $$ \\times $$ 10<sup>2</sup> J"}, {"identifier": "D", "content": "3.39 $$ \\times $$ 10<sup>3</sup> J"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263369/exam_images/ajrqexexn67nnrokgcbn.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 9th January Evening Slot Physics - Alternating Current Question 140 English Explanation">
<br><b... | mcq | jee-main-2019-online-9th-january-evening-slot | 8,845 |
daWQCwZz5379riDFnM7k9k2k5dolaqg | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A LCR circuit behaves like a damped harmonic oscillator. Comparing it with a physical spring-mass damped oscillator having damping constant 'b', the correct equivalence would be: | [{"identifier": "A", "content": "L $$ \\leftrightarrow $$ k, C $$ \\leftrightarrow $$ b, R $$ \\leftrightarrow $$ m"}, {"identifier": "B", "content": "L $$ \\leftrightarrow $$ m, C $$ \\leftrightarrow $$ k, R $$ \\leftrightarrow $$ b"}, {"identifier": "C", "content": "L $$ \\leftrightarrow $$ m, C $$ \\leftrightarrow $... | ["C"] | null | For spring mass damped oscillator
<br><br>ma = - kx - bv
<br><br>$$ \Rightarrow $$ ma + kx + bv = 0
<br><br>$$ \Rightarrow $$ $$m{{{d^2}x} \over {d{t^2}}}$$ + b$${{dx} \over {dt}}$$ + kx = 0 ....(1)
<br><br>For LCR circuit
<br><br>L$${{di} \over {dt}}$$ + iR + $${q \over C}$$ = 0
<br><br>$$ \Rightarrow $$ L$${{{d^2}q} ... | mcq | jee-main-2020-online-7th-january-morning-slot | 8,846 |
qefVorS2uEXC7q9z647k9k2k5l5yars | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In LC circuit the inductance L = 40 mH and
<br/>capacitance C = 100 $$\mu $$F. If a voltage
<br/>V(t) = 10sin(314t) is applied to the circuit, the
<br/>current in the circuit is given as : | [{"identifier": "A", "content": "0.52 cos 314 t"}, {"identifier": "B", "content": "5.2 cos 314 t"}, {"identifier": "C", "content": "0.52 sin 314 t"}, {"identifier": "D", "content": "10 cos 314 t"}] | ["A"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263802/exam_images/i0j0zbm9qpkivxppcwe2.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 9th January Evening Slot Physics - Alternating Current Question 129 English Explanation">
<br><br>... | mcq | jee-main-2020-online-9th-january-evening-slot | 8,847 |
VEEt9YXhR16lconjbojgy2xukexwg41v | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | An inductance coil has a reactance of 100 $$\Omega $$.
When an AC signal of frequency 1000 Hz is
applied to the coil, the applied voltage leads
the current by 45<sup>o</sup>. The self-inductance of the
coil is | [{"identifier": "A", "content": "6.7 $$ \\times $$ 10<sup>\u20137</sup> H"}, {"identifier": "B", "content": "1.1 $$ \\times $$ 10<sup>\u20131</sup> H"}, {"identifier": "C", "content": "5.5 $$ \\times $$ 10<sup>\u20135</sup> H"}, {"identifier": "D", "content": "1.1 $$ \\times $$ 10<sup>\u20132</sup> H"}] | ["D"] | null | L-R circuit :
<br><br>tan 45<sup>o</sup> = $${{{X_L}} \over R}$$
<br><br>$$ \Rightarrow $$ 1 = $${{{X_L}} \over R}$$
<br><br>$$ \Rightarrow $$ X<sub>L</sub> = R
<br><br>Now Z = $$\sqrt {{R^2} + X_L^2} $$
<br><br>or Z = $$\sqrt {X_L^2 + X_L^2} = \sqrt {2X_L^2} $$ = $$\sqrt 2 {X_L}$$
<br><br>$$ \Rightarrow $$ 100 = $$\s... | mcq | jee-main-2020-online-2nd-september-evening-slot | 8,848 |
44MzeCC1nuPxzqf3q5jgy2xukf15pezu | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A 750 Hz, 20 V (rms) source is connected to a
resistance of 100 $$\Omega $$, an inductance of 0.1803 H
and a capacitance of 10 $$\mu $$F all in series. The
time in which the resistance (heat capacity
2 J/<sup>o</sup>C) will get heated by 10<sup>o</sup>C. (assume no loss
of heat to the surroudnings) is close to : | [{"identifier": "A", "content": "348 s"}, {"identifier": "B", "content": "418 s"}, {"identifier": "C", "content": "245 s"}, {"identifier": "D", "content": "365 s"}] | ["A"] | null | f = 750 Hz, V<sub>rms</sub> = 20 V,
<br><br>R = 100 $$\Omega $$, L = 0.1803 H,
<br><br>C = 10$$\mu $$ F, S = 2 J/°C
<br><br>|Z| = $$\sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} $$
<br><br>= $$\sqrt {{R^2} + {{\left( {\omega L - {1 \over {\omega C}}} \right)}^2}} $$
<br><br>= $$\sqrt {{R^2} + {{\left( {2\pi fL -... | mcq | jee-main-2020-online-3rd-september-morning-slot | 8,849 |
ftYWCKEbBX1ANDU6Gijgy2xukg0kk5t5 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In a series LR circuit, power of 400W is dissipated from a source of 250 V, 50 Hz. The power factor
of the circuit is 0.8. In order to bring the power factor to unity, a capacitor of value C is added in
series to the L and R. Taking the value of C as $$\left( {{n \over {3\pi }}} \right)$$ $$\mu $$F, then value of n is... | [] | null | 400 | Given, power factor of LR circuit,
<br><br>cos $$\phi $$ = 0.8 = $${R \over {\sqrt {{R^2} + X_L^2} }}$$ = $${R \over Z}$$
<br><br>We know,
<br>Power, P = $${{V_{rms}^2} \over {{Z^2}}} \times R$$
<br><br>$$ \Rightarrow $$ 400 = $${{{{\left( {250} \right)}^2} \times 0.8Z} \over {{Z^2}}}$$
<br><br>$$ \Rightarrow $$ Z = 1... | integer | jee-main-2020-online-6th-september-evening-slot | 8,850 |
XypluqVNzJh3Dh6K3o1klroxzmw | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A series L-C-R circuit is designed to resonate at an angular frequency $$\omega$$<sub>0</sub> = 10<sup>5</sup> rad/s. The circuit draws 16W power from 120V source at resonance. The value of resistance 'R' in the circuit is _________ $$\Omega$$. | [] | null | 900 | Given, angular frequency at resonance, $$\omega$$<sub>0</sub> = 10<sup>5</sup> rads<sup>$$-$$1</sup><br/><br/>Power drawn from circuit, P = 16 W<br/><br/>and supply voltage, V = 120 V<br/><br/>Let resistance of circuit = R.<br/><br/>As, $$P = {V^2}/R$$<br/><br/>$$ \Rightarrow R = {V^2}/P = {{120 \times 100} \over {16}}... | integer | jee-main-2021-online-24th-february-evening-slot | 8,851 |
hF94OJCRU38FHo9ewG1klrx1qb1 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | The angular frequency of alternating current in a L-C-R circuit is 100 rad/s. The components connected are shown in the figure. Find the value of inductance of the coil and capacity of condenser.<br/><br/>
<img src="data:image/png;base64,UklGRuwNAABXRUJQVlA4IOANAABQSgCdASpVAacAPm02lkikIqIhInK6yIANiWlu4WeRG/OZ8U/yP8aPAz... | [{"identifier": "A", "content": "0.8 H and 150 $$\\mu$$F"}, {"identifier": "B", "content": "0.8 H and 250 $$\\mu$$F"}, {"identifier": "C", "content": "1.33 H and 150 $$\\mu$$F"}, {"identifier": "D", "content": "1.33 H and 250 $$\\mu$$F"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266685/exam_images/njdiyzkeri6bjruz5cqc.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 25th February Morning Shift Physics - Alternating Current Question 121 English Explanation">
<br>S... | mcq | jee-main-2021-online-25th-february-morning-slot | 8,852 |
nH2aE2MWNH3oD5egKk1klrymbpp | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | The current (i) at time t = 0 and t = $$\infty $$ respectively for the given circuit is :<br/><br/><img src="data:image/png;base64,UklGRkAMAABXRUJQVlA4IDQMAABwRACdASobAb8APm0ylUikIqIhInILCIANiWlu3WBpKb+kH8k/ITwT/sf5ReIz5b+ifkhnK/qP9o/k39J/2/9o9bv71/Rv2V/ID3J9kfxF7AXpj+zfx3+b/6r+scLxA77AXav/H/yn++f8z/Iecl+r/wD+Af6jyl/2v... | [{"identifier": "A", "content": "$${{18E} \\over {55}},{{5E} \\over {18}}$$"}, {"identifier": "B", "content": "$${{5E} \\over {18}},{{18E} \\over {55}}$$"}, {"identifier": "C", "content": "$${{5E} \\over {18}},{{10E} \\over {33}}$$"}, {"identifier": "D", "content": "$${{10E} \\over {33}},{{5E} \\over {18}}$$"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267557/exam_images/f32xtb0jhvl9lfixcqhm.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 25th February Morning Shift Physics - Alternating Current Question 119 English Explanation 1"><br>... | mcq | jee-main-2021-online-25th-february-morning-slot | 8,853 |
rZe3uUOxwxTFT9b0EE1klrz9r98 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A transmitting station releases waves of wavelength 960 m. A capacitor of 2.56 $$\mu$$F is used in the resonant circuit. The self inductance of coil necessary for resonance is __________ $$\times$$ 10<sup>$$-$$8</sup> H. | [] | null | 10 | $$\lambda$$ = 960 m<br><br>C = 2.56 $$\mu$$F = 2.56 $$\times$$ 10<sup>$$-$$6</sup><sup></sup> F<br><br>c = 3 $$\times$$ 10<sup>8</sup> m/s<br><br>L = ?<br><br>Now at resonance, $${\omega _0} = {1 \over {\sqrt {LC} }}$$<br><br>[Resonant frequency]<br><br>$$2\pi {f_0} = {1 \over {\sqrt {LC} }}$$<br><br>On substituting $$... | integer | jee-main-2021-online-25th-february-morning-slot | 8,854 |
8OGOKavxUO7OfLwZaI1klt2p82o | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | Match List I with List II.<br/><br/><table>
<thead>
<tr>
<th></th>
<th>List I</th>
<th></th>
<th>List II</th>
</tr>
</thead>
<tbody>
<tr>
<td>(a)</td>
<td>Rectifier</td>
<td>(i)</td>
<td>Used either for stepping up or stepping down the a.c. voltage</td>
</tr>
<tr>
<td>(b)</td>
<td>Stabilizer</td>
<td>(ii)</td>
<td>Used... | [{"identifier": "A", "content": "(a)-(ii), (b)-(iv), (c)-(i), (d)-(iii)"}, {"identifier": "B", "content": "(a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)"}, {"identifier": "C", "content": "(a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)"}, {"identifier": "D", "content": "(a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)"}] | ["A"] | null | (a) Rectifier : used to convert a a.c. voltage into d.c. voltage.<br><br>(b) Stabilizer : used for constant output voltage even when the input voltage or load current change<br><br>(c) Transformer : used either for stepping up or stepping down the a.c. voltage.<br><br>(d) Filter : used to remove any ripple in the recti... | mcq | jee-main-2021-online-25th-february-evening-slot | 8,856 |
SqC8mex5Z0YEMvwrZ81kltjmg8c | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In a series LCR resonant circuit, the quality factor is measured as 100. If the inductance is increased by two fold and resistance is decreased by two fold, then the quality factor after this change will be __________. | [] | null | 283 | Quality factor = $${{{X_L}} \over R} = {{\omega L} \over R}$$<br><br>$$Q = {1 \over {\sqrt {LC} }}{L \over R}$$<br><br>$$Q = \left( {{1 \over {\sqrt C }}} \right){{\sqrt L } \over R}$$<br><br>$$Q = {{XL} \over R} = {{\omega L} \over R} = {1 \over {\sqrt {LC} }}{L \over R} = {1 \over R}{{\sqrt L } \over {\sqrt C }}$$<br... | integer | jee-main-2021-online-26th-february-morning-slot | 8,858 |
8vt2YCngI2nQxFRMNo1klukf8rq | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | Find the peak current and resonant frequency of the following circuit (as shown in figure).<br/><br/><img src="data:image/png;base64,UklGRgQNAABXRUJQVlA4IPgMAAAQRgCdASoMAa4APm00lUikIqIhI1Q7IIANiWlu/GdPPG208Rfzv8WvA/+zflJ59/hnyf9W/on7DfvDl4/vP8s/qH+x9Iv7n/MP6N/jP7J6Q+1z909QL1b/bf59+0/9m9JH9v/jX9V79Cr/oBe0/0n/Of0b+wf6fyd... | [{"identifier": "A", "content": "2 A and 100 Hz"}, {"identifier": "B", "content": "2 A and 50 Hz"}, {"identifier": "C", "content": "0.2 A and 100 Hz"}, {"identifier": "D", "content": "0.2 A and 50 Hz"}] | ["D"] | null | We know, z = $$\sqrt {{{({x_L} - {x_C})}^2} + {R^2}} $$<br><br>$${x_L} = {\omega _L} = 100 \times 100 \times {10^{ - 3}} = 10\Omega $$<br><br>$${x_C} = {1 \over {{\omega _C}}} = {1 \over {100 \times 100 \times {{10}^{ - 6}}}} = 10\Omega $$<br><br>$$ \therefore $$ $$z = \sqrt {{{(10 - 100)}^2} + {R^2}} = \sqrt {{{90}^2... | mcq | jee-main-2021-online-26th-february-evening-slot | 8,859 |
sllZCVBCjsoJIpQFaK1kmhpj37e | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A sinusoidal voltage of peak value 250 V is applied to a series LCR circuit, in which R = 8$$\Omega$$, L = 24 mH and C = 60 $$\mu$$F. The value of power dissipated at resonant condition is 'x' kW. The value of x to the nearest integer is ____________. | [] | null | 4 | At resonance power (P)<br><br>$$P = {{{{({V_{rms}})}^2}} \over R}$$<br><br>$$ \therefore $$ $$P = {{{{(250/\sqrt 2 )}^2}} \over 8}$$<br><br>$$ \Rightarrow $$ P = 3906.25 w<br><br>$$ \Rightarrow $$ P $$ \cong $$ 4 Kw | integer | jee-main-2021-online-16th-march-morning-shift | 8,860 |
VF3xHLQMu4uzx3s5Dr1kmkrhkv9 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In a series LCR resonance circuit, if we change the resistance only, from a lower to higher value : | [{"identifier": "A", "content": "The bandwidth of resonance circuit will increase."}, {"identifier": "B", "content": "The resonance frequency will increase."}, {"identifier": "C", "content": "The quality factor will increase."}, {"identifier": "D", "content": "The quality factor and the resonance frequency will remain ... | ["A"] | null | $${\omega } = {1 \over {\sqrt {LC} }}$$
<br><br>$$ \Rightarrow $$ 2$$\pi $$f = $${1 \over {\sqrt {LC} }}$$
<br><br>$$ \Rightarrow $$ f = $${1 \over {2\pi \sqrt {LC} }}$$
<br><br>f does not depends on resistance(R).
<br><br>Quality factor, $$Q = {{\omega L} \over R}$$
<br><br>$$ \Rightarrow $$ $$Q \propto {1 \over R}$$
... | mcq | jee-main-2021-online-18th-march-morning-shift | 8,864 |
gQhRMzmtxQbTm9HQHU1kmkrijd1 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | An AC source rated 220 V, 50 Hz is connected to a resistor. The time taken by the current to change from its maximum to the rms value is : | [{"identifier": "A", "content": "2.5 ms"}, {"identifier": "B", "content": "25 ms"}, {"identifier": "C", "content": "2.5 s"}, {"identifier": "D", "content": "0.25 ms"}] | ["A"] | null | $$I = {I_0}\sin \omega t $$
<br><br>$$ \Rightarrow $$ $${{{I_M}} \over {\sqrt 2 }} = {I_M}\sin \omega t$$
<br><br>$$ \Rightarrow $$ $$\omega t = {\pi \over 4}$$
<br><br>$$ \Rightarrow $$ $$t = {\pi \over {4\omega }}$$ $$ = {\pi \over {4\left( {2\pi f} \right)}}$$
<br><br>$$ \Rightarrow $$ t = $${1 \over {8 \times 30... | mcq | jee-main-2021-online-18th-march-morning-shift | 8,865 |
drcpQeI6nE0zuYmBRD1kmlvm9lc | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In a series LCR circuit, the inductive reactance (X<sub>L</sub>) is 10$$\Omega$$ and the capacitive reactance (X<sub>C</sub>) is 4$$\Omega$$. The resistance (R) in the circuit is 6$$\Omega$$. The power factor of the circuit is : | [{"identifier": "A", "content": "$${1 \\over 2}$$"}, {"identifier": "B", "content": "$${{\\sqrt 3 } \\over 2}$$"}, {"identifier": "C", "content": "$${1 \\over {\\sqrt 2 }}$$"}, {"identifier": "D", "content": "$${1 \\over {2\\sqrt 2 }}$$"}] | ["C"] | null | Given :<br><br>X<sub>L</sub> = 10$$\Omega$$<br><br>X<sub>C</sub> = 4$$\Omega$$<br><br>R = 6$$\Omega$$<br><br>$$ \therefore $$ Power factor = cos$$\theta$$ = $${R \over Z}$$<br><br>$$ = {R \over {\sqrt {{R^2} + {{({X_L} - {X_C})}^2}} }}$$<br><br>$$ = {6 \over {\sqrt {{6^2} + {{(10 - 4)}^2}} }}$$<br><br>$$ = {6 \over {6\... | mcq | jee-main-2021-online-18th-march-evening-shift | 8,866 |
1krpnqvvj | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | AC voltage V(t) = 20 sin$$\omega$$t of frequency 50 Hz is applied to a parallel plate capacitor. The separation between the plates is 2 mm and the area is 1 m<sup>2</sup>. The amplitude of the oscillating displacement current for the applied AC voltage is _________. [Take $$\varepsilon $$<sub>0</sub> = 8.85 $$\times$$ ... | [{"identifier": "A", "content": "55.58 $$\\mu$$A"}, {"identifier": "B", "content": "21.14 $$\\mu$$A"}, {"identifier": "C", "content": "27.79 $$\\mu$$A"}, {"identifier": "D", "content": "83.37 $$\\mu$$A"}] | ["C"] | null | Given,<br/><br/>AC voltage, V(t) = 20 sin $$\omega$$t volt.<br/><br/>Frequency, f = 50Hz<br/><br/>Separation between the plates, d = 2 mm = 2 $$\times$$ 10<sup>$$-$$3</sup> m<br/><br/>Area, A = 1 m<sup>2</sup><br/><br/>As, $$C = {{{\varepsilon _0}A} \over d}$$<br/><br/>where, $${{\varepsilon _0}}$$ = absolute electrica... | mcq | jee-main-2021-online-20th-july-morning-shift | 8,867 |
1krpqcms4 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In an LCR series circuit, an inductor 30 mH and a resistor 1 $$\Omega$$ are connected to an AC source of angular frequency 300 rad/s. The value of capacitance for which, the current leads the voltage by 45$$^\circ$$ is $${1 \over x} \times {10^{ - 3}}$$ F. Then the value of x is ____________. | [] | null | 3 | Given,<br/><br/>Inductance, L = 30 mH<br/><br/>Resistance, R = 1 $$\Omega$$<br/><br/>Angular frequency, $$\omega$$ = 300 rad/s<br/><br/>We know that in L-C-R circuit, $$\tan \phi = {{{X_C} - {X_L}} \over R}$$<br/><br/>where, $$\phi$$ = phase angle = 45$$^\circ$$<br/><br/>X<sub>C</sub> = capacitive reactance = $${1 \ov... | integer | jee-main-2021-online-20th-july-morning-shift | 8,868 |
1krqbc8pz | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | For a series LCR circuit with R = 100 $$\Omega$$, L = 0.5 mH and C = 0.1 pF connected across 220V$$-$$50 Hz AC supply, the phase angle between current and supplied voltage and the nature of the circuit is : | [{"identifier": "A", "content": "0$$^\\circ$$, resistive circuit"}, {"identifier": "B", "content": "$$ \\approx $$ 90$$^\\circ$$, predominantly inductive circuit"}, {"identifier": "C", "content": "0$$^\\circ$$, resonance circuit"}, {"identifier": "D", "content": "$$ \\approx $$ 90$$^\\circ$$, predominantly capacitive c... | ["D"] | null | R = 100$$\Omega$$<br><br>$${X_L} = \omega L = 50\pi \times {10^{ - 3}}$$<br><br>$${X_C} = {1 \over {\omega C}} = {{{{10}^{11}}} \over {100\pi }}$$<br><br>$${X_C} > > {X_L}$$<br><br>& $$\left| {{X_C} - {X_L}} \right| > > R$$ | mcq | jee-main-2021-online-20th-july-evening-shift | 8,869 |
1krqf6xhj | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A series LCR circuit of R = 5$$\Omega$$, L = 20 mH and C = 0.5 $$\mu$$F is connected across an AC supply of 250 V, having variable frequency. The power dissipated at resonance condition is ______________ $$\times$$ 10<sup>2</sup> W. | [] | null | 125 | X<sub>L</sub> = X<sub>C</sub> (due to resonance)<br><br>Z = R so $${i_{rms}} = {V \over Z} = {V \over R}$$<br><br>$${{{V^2}} \over R} = {{250 \times 250} \over 5} = 125 \times {10^2}W$$ | integer | jee-main-2021-online-20th-july-evening-shift | 8,870 |
1krstzh54 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In a circuit consisting of a capacitance and a generator with alternating emf E<sub>g</sub> = E<sub>g<sub>0</sub></sub> sin$$\omega$$t, V<sub>C</sub> and I<sub>C</sub> are the voltage and current. Correct phasor diagram for such circuit is<br/><br/><img src="data:image/png;base64,UklGRpwIAABXRUJQVlA4IJAIAAAwQACdASpAAcM... | [{"identifier": "A", "content": "<img src=\"https://res.cloudinary.com/dckxllbjy/image/upload/v1734267385/exam_images/cfqauv3myar4llkqpn0b.webp\" style=\"max-width: 100%;height: auto;display: block;margin: 0 auto;\" loading=\"lazy\" alt=\"JEE Main 2021 (Online) 22th July Evening Shift Physics - Alternating Current Ques... | ["C"] | null | In capacitor, current lead voltage by $$\pi\over2$$ | mcq | jee-main-2021-online-22th-july-evening-shift | 8,871 |
1krsv7oe0 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | Match List - I with List - II<br/><br/><table>
<thead>
<tr>
<th></th>
<th>List - I</th>
<th></th>
<th>List - II</th>
</tr>
</thead>
<tbody>
<tr>
<td>(a)</td>
<td>$$\omega L > {1 \over {\omega C}}$$</td>
<td>(i)</td>
<td>Current is in phase with emf</td>
</tr>
<tr>
<td>(b)</td>
<td>$$\omega L = {1 \over {\omega C}}$$... | [{"identifier": "A", "content": "a(ii), b(i), c(iv), d(iii)"}, {"identifier": "B", "content": "a(ii), b(i), c(iii), d(iv)"}, {"identifier": "C", "content": "a(iii), b(i), c(iv), d(ii)"}, {"identifier": "D", "content": "a(iv), b(iii), c(ii), d(i)"}] | ["A"] | null | $$\omega L = {1 \over {\omega C}},{X_L} = {X_C}$$<br><br>So current in phase with EMF<br><br>At resonance, current have maximum value. | mcq | jee-main-2021-online-22th-july-evening-shift | 8,872 |
1krw8vbh1 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A 10 $$\Omega$$ resistance is connected across 220V $$-$$ 50 Hz AC supply. The time taken by the current to change from its maximum value to the rms value is : | [{"identifier": "A", "content": "2.5 ms"}, {"identifier": "B", "content": "1.5 ms"}, {"identifier": "C", "content": "3.0 ms"}, {"identifier": "D", "content": "4.5 ms"}] | ["A"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266733/exam_images/sh9bt7r6jxkdnjuusuxf.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264606/exam_images/qxtg6qdeajprayist6jx.webp"><img src="https://res.c... | mcq | jee-main-2021-online-25th-july-evening-shift | 8,873 |
1krwcgvnw | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | Two circuits are shown in the figure (a) & (b). At a frequency of ____________ rad/s the average power dissipated in one cycle will be same in both the circuits.<br/><br/><img src="data:image/png;base64,UklGRr4YAABXRUJQVlA4ILIYAADQhwCdASpsAuEAPm02lkkkIqKhIbEaYIANiWlu/B4IAtgbbVG39Fv45+M/f7/ZfyT89fxL5d+tf1n9mP69/5v9V... | [] | null | 500 | For figure (a)<br><br>$${P_{avg}} = {{v_{rms}^2} \over R}$$<br><br>$${{v_{rms}^2} \over {{Z^2}}} \times R = {{v_{rms}^2} \over R} \times 1$$<br><br>$${R^2} = {Z^2}$$<br><br>$$25 = {\left( {\sqrt {{{({x_C} - {x_L})}^2} + {5^2}} } \right)^2}$$<br><br>$$ = 25{({x_C} - {x_L})^2} + 25$$<br><br>$${x_C} = {x_L} \Rightarrow {1... | integer | jee-main-2021-online-25th-july-evening-shift | 8,874 |
1kryx1618 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A 0.07 H inductor and a 12$$\Omega$$ resistor are connected in series to a 220V, 50 Hz ac source. The approximate current in the circuit and the phase angle between current and source voltage are respectively. [Take $$\pi$$ as $${{22} \over 7}$$] | [{"identifier": "A", "content": "8.8 A and $${\\tan ^{ - 1}}\\left( {{{11} \\over 6}} \\right)$$"}, {"identifier": "B", "content": "88 A and $${\\tan ^{ - 1}}\\left( {{{11} \\over 6}} \\right)$$"}, {"identifier": "C", "content": "0.88 A and $${\\tan ^{ - 1}}\\left( {{{11} \\over 6}} \\right)$$"}, {"identifier": "D", "c... | ["A"] | null | $$\phi = {\tan ^{ - 1}}\left( {{{{X_L}} \over R}} \right)$$<br><br>$${X_L} = \omega L$$<br><br>$${X_L} = 2 \times {{22} \over 7} \times 50 \times 0.07 = 22\Omega $$<br><br>$$\phi = {\tan ^{ - 1}}\left( {{{22} \over {12}}} \right)$$<br><br>$$R = 12\Omega $$<br><br>$$\phi = {\tan ^{ - 1}}\left( {{{11} \over 6}} \right... | mcq | jee-main-2021-online-27th-july-morning-shift | 8,875 |
1ks17s7bk | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A 100$$\Omega$$ resistance, a 0.1 $$\mu$$F capacitor and an inductor are connected in series across a 250 V supply at variable frequency. Calculate the value of inductance of inductor at which resonance will occur. Given that the resonant frequency is 60 Hz. | [{"identifier": "A", "content": "0.70 H"}, {"identifier": "B", "content": "70.3 mH"}, {"identifier": "C", "content": "7.03 $$\\times$$ 10<sup>$$-$$5</sup> H"}, {"identifier": "D", "content": "70.3 H"}] | ["D"] | null | C = 0.1 $$\mu$$F = 10<sup>$$-$$7</sup> F<br><br>Resonant frequency = 60 Hz.<br><br>$${\omega _0} = {1 \over {\sqrt {LC} }}$$<br><br>$$2\pi {f_0} = {1 \over {\sqrt {LC} }} \Rightarrow L = {1 \over {4{\pi ^2}f_0^2C}}$$<br><br>by putting values $$L \simeq 70.3$$ Hz. | mcq | jee-main-2021-online-27th-july-evening-shift | 8,876 |
1ktbriwma | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In the given circuit the AC source has $$\omega$$ = 100 rad s<sup>-1</sup>. Considering the inductor and capacitor to be ideal, what will be the current I flowing through the circuit? <br/><br/><img src="data:image/png;base64,UklGRjISAABXRUJQVlA4ICYSAAAQaQCdASqIAQ8BPm0ylUkkIqIhIVMK4IANiWlu6B/I6BNWuqNv6LfzX8gPBz+ofkr4i/... | [{"identifier": "A", "content": "5.9 A"}, {"identifier": "B", "content": "3.16 A"}, {"identifier": "C", "content": "0.94 A"}, {"identifier": "D", "content": "6 A"}] | ["B"] | null | $${Z_C} = \sqrt {{{\left( {{1 \over {\omega C}}} \right)}^2} + {R^2}} $$<br><br>$$ = \sqrt {{{\left( {{1 \over {100 \times 100 \times {{10}^{ - 6}}}}} \right)}^2} + {{100}^2}} $$<br><br>$${Z_C} = \sqrt {{{(100)}^2} + {{(100)}^2}} $$<br><br>$$ = 100\sqrt 2 $$<br><br>$${Z_L} = \sqrt {{{(\omega L)}^2} + {R^2}} $$<br><br>$... | mcq | jee-main-2021-online-26th-august-evening-shift | 8,878 |
1ktfodray | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | An ac circuit has an inductor and a resistor resistance R in series, such that X<sub>L</sub> = 3R. Now, a capacitor is added in series such that X<sub>C</sub> = 2R. The ratio of new power factor with the old power factor of the circuit is $$\sqrt 5 :x$$. The value of x is ___________. | [] | null | 1 | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267580/exam_images/e4vsle7asn82li8wjstj.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263363/exam_images/nbwuqu1jzyrth9bhnonm.webp"><img src="https://res.c... | integer | jee-main-2021-online-27th-august-evening-shift | 8,880 |
1kth5cv9x | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | In an ac circuit, an inductor, a capacitor and a resistor are connected in series with X<sub>L</sub> = R = X<sub>C</sub>. Impedance of this circuit is : | [{"identifier": "A", "content": "2R<sup>2</sup>"}, {"identifier": "B", "content": "Zero"}, {"identifier": "C", "content": "R"}, {"identifier": "D", "content": "R$$\\sqrt 2 $$"}] | ["C"] | null | $$Z = \sqrt {{{({X_L} - {X_C})}^2} + {R^2}} = R$$ $$\because$$ X<sub>L</sub> = X<sub>C</sub><br><br>Option (c) | mcq | jee-main-2021-online-31st-august-morning-shift | 8,881 |
1ktjqyc7e | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | At very high frequencies, the effective impendence of the given circuit will be ________________ $$\Omega$$.<br/><br/><img src="data:image/png;base64,UklGRjQYAABXRUJQVlA4ICgYAADQiQCdASoGAgwBPm0ylUkkIqIhIZFK8IANiWlu/F+4hQonZ10/pB/O/xf8Ev6/+T3iV+b/qn9Q/ZX+2f+r3FvzPpRv2r+V+qf80+qH1z+Qf0L/afl793fzD+w/yL+X/2f8wfbP4RfxHqEen/... | [] | null | 2 | X<sub>L</sub> = 2$$\pi$$fL<br><br>f is very large<br><br>$$\therefore$$ X<sub>L</sub> is very large hence open circuit.<br><br>$${X_C} = {1 \over {2\pi fC}}$$<br><br>f is very large.<br><br>$$\therefore$$ X<sub>C</sub> is very small, hence short circuit.<br><br>Final circuit<br><br><img src="https://res.cloudinary.com/... | integer | jee-main-2021-online-31st-august-evening-shift | 8,882 |
1l547cca4 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>For a series LCR circuit, I vs $$\omega$$ curve is shown :</p>
<p>(a) To the left of $$\omega$$<sub>r</sub>, the circuit is mainly capacitive.</p>
<p>(b) To the left of $$\omega$$<sub>r</sub>, the circuit is mainly inductive.</p>
<p>(c) At $$\omega$$<sub>r</sub>, impedance of the circuit is equal to the resistance o... | [{"identifier": "A", "content": "(a) and (d) only."}, {"identifier": "B", "content": "(b) and (d) only."}, {"identifier": "C", "content": "(a) and (c) only."}, {"identifier": "D", "content": "(b) and (c) only."}] | ["C"] | null | <p>We know that $${X_C} = {1 \over {\omega C}}$$ and $${X_L} = \omega L$$</p>
<p>Also, at $$\omega = {\omega _r}:{X_L} = {X_C}$$</p>
<p>$$\Rightarrow$$ For $$\omega < {\omega _r}$$ : capacitive</p>
<p>and $$\omega = {\omega _r}:z = \sqrt {{R^2} + {{({X_L} - {X_C})}^2}} = R$$</p> | mcq | jee-main-2022-online-29th-june-morning-shift | 8,883 |
1l54x17ph | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>An inductor of 0.5 mH, a capacitor of 200 $$\mu$$F and a resistor of 2 $$\Omega$$ are connected in series with a 220 V ac source. If the current is in phase with the emf, the frequency of ac source will be ____________ $$\times$$ 10<sup>2</sup> Hz.</p> | [] | null | 5 | <p>Current will be in phase with emf when</p>
<p>$$\omega L = {1 \over {\omega C}}$$</p>
<p>$$ \Rightarrow \omega = {1 \over {\sqrt {LC} }} = {1 \over {\sqrt {5 \times {{10}^{ - 4}} \times 2 \times {{10}^{ - 4}}} }}$$</p>
<p>$$ \Rightarrow \omega = {{{{10}^4}} \over {\sqrt {10} }}$$ rad/s</p>
<p>$$ \Rightarrow f = {1... | integer | jee-main-2022-online-29th-june-evening-shift | 8,884 |
1l55m1te8 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>In the given circuit, the magnitude of V<sub>L</sub> and V<sub>C</sub> are twice that of V<sub>R</sub>. Given that f = 50 Hz, the inductance of the coil is $${1 \over {K\pi }}$$ mH. The value of K is ____________.</p>
<p> <img src="data:image/png;base64,UklGRsANAABXRUJQVlA4ILQNAAAw8ACdASoAA6QCP4HA3GW2MK2nIZOJEsAwCWl... | [] | null | 0 | <p>$${V_L} = 2{V_R}$$</p>
<p>So $$\omega Li = 2\,Ri$$</p>
<p>$$ \Rightarrow L = {{2R} \over \omega } = {{2 \times 5} \over {2\pi \times 50}} = {1 \over {10\pi }}H = {{100} \over \pi }H$$</p>
<p>So $$k = {1 \over {100}} \simeq 0$$</p> | integer | jee-main-2022-online-28th-june-evening-shift | 8,885 |
1l56a70y8 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>An AC source is connected to an inductance of 100 mH, a capacitance of 100 $$\mu$$F and a resistance of 120 $$\Omega$$ as shown in figure. The time in which the resistance having a thermal capacity 2 J/$$^\circ$$C will get heated by 16$$^\circ$$C is _____________ s.</p>
<p><img src="data:image/png;base64,UklGRjYMAAB... | [] | null | 15 | <p>L = 100 $$\times$$ 10<sup>$$-$$3</sup> H</p>
<p>C = 100 $$\times$$ 10<sup>$$-$$6</sup> F</p>
<p>R = 120 $$\Omega$$</p>
<p>$$\omega$$L = 10 $$\Omega$$</p>
<p>$${1 \over {\omega C}} = {1 \over {{{10}^4} \times {{10}^{ - 6}}}} = 100\,\Omega $$</p>
<p>$$\Rightarrow$$ X<sub>C</sub> $$-$$ X<sub>L</sub> = 90 $$\Omega$$</p>... | integer | jee-main-2022-online-28th-june-morning-shift | 8,886 |
1l56afv0x | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>A telegraph line of length 100 km has a capacity of 0.01 $$\mu$$F/km and it carries an alternating current at 0.5 kilo cycle per second. If minimum impedance is required, then the value of the inductance that needs to be introduced in series is _____________ mH. (if $$\pi$$ = $$\sqrt{10}$$)</p> | [] | null | 100 | <p>Total capacitance = 0.01 $$\times$$ 100 = 1 $$\mu$$F</p>
<p>$$\omega$$ = 500 $$\times$$ 2$$\pi$$ = 1000$$\pi$$ rad/s</p>
<p>$$\omega L = {1 \over {\omega C}}$$</p>
<p>$$ \Rightarrow L = {1 \over {{\omega ^2}C}} = {1 \over {{{10}^6}{\pi ^2} \times {{10}^{ - 6}}}} = {1 \over {10}}H$$ = 100 mH</p> | integer | jee-main-2022-online-28th-june-morning-shift | 8,887 |
1l57qnm6k | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>A 220 V, 50 Hz AC source is connected to a 25 V, 5 W lamp and an additional resistance R in series (as shown in figure) to run the lamp at its peak brightness, then the value of R (in ohm) will be _____________.</p>
<p> <img src="data:image/png;base64,UklGRnAMAABXRUJQVlA4IGQMAABwxQCdASoAA/4BP4HA22U2MS2nIZcZWsAwCWlu7... | [] | null | 975 | <p>$${R_b} = {{{{(25)}^2}} \over 5} = 125\,\Omega $$</p>
<p>$${I_{rms}} = \sqrt {{5 \over {125}}} = {1 \over 5}A$$</p>
<p>$$ \Rightarrow {{220} \over {R + 125}} = {1 \over 5}$$</p>
<p>$$ \Rightarrow R = 1100 - 125$$</p>
<p>$$ = 975\,\Omega $$</p> | integer | jee-main-2022-online-27th-june-morning-shift | 8,888 |
1l58d6736 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>A 110 V, 50 Hz, AC source is connected in the circuit (as shown in figure). The current through the resistance 55 $$\Omega$$, at resonance in the circuit, will be __________ A.</p>
<p><img src="data:image/png;base64,UklGRqINAABXRUJQVlA4IJYNAABQ6wCdASoAA1ICP4HA3GS2MTunIfXJY3AwCWlu4XMjqmNwu56u/3frQvG9+vcXiQYkKv/nxGTn/... | [] | null | 0 | $$
\frac{1}{Z}=\sqrt{\left(\frac{1}{X_L}-\frac{1}{X_C}\right)^2}
$$
<br/><br/>At resonance, $\mathrm{X}_{\mathrm{L}}=\mathrm{X}_{\mathrm{C}} \& \mathrm{Z} \rightarrow \infty$
<br/><br/>$\therefore \mathrm{Z}_{\text {total circuit }} \rightarrow \infty$
<br/><br/>i.e, $\mathrm{I}=0$ | integer | jee-main-2022-online-26th-june-morning-shift | 8,889 |
1l59p78c8 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>A sinusoidal voltage V(t) = 210 sin 3000 t volt is applied to a series LCR circuit in which L = 10 mH, C = 25 $$\mu$$F and R = 100 $$\Omega$$. The phase difference ($$\Phi $$) between the applied voltage and resultant current will be :</p> | [{"identifier": "A", "content": "tan<sup>$$-$$1</sup>(0.17)"}, {"identifier": "B", "content": "tan<sup>$$-$$1</sup>(9.46)"}, {"identifier": "C", "content": "tan<sup>$$-$$1</sup>(0.30)"}, {"identifier": "D", "content": "tan<sup>$$-$$1</sup>(13.33)"}] | ["A"] | null | <p>$${X_L} = 3000 \times 10 \times {10^{ - 3}} = 30\,\Omega $$</p>
<p>$${X_C} = {1 \over {3000 \times 25}} \times {10^6} = {{40} \over 3}\,\Omega $$</p>
<p>So $${X_L} - {X_C} = 30 - {{40} \over 3} = {{50} \over 3}\,\Omega $$</p>
<p>$$\tan \theta = {{{X_L} - {X_C}} \over R} = {{50/3} \over {100}} = {1 \over 6}$$</p>
<p... | mcq | jee-main-2022-online-25th-june-evening-shift | 8,890 |
1l5aknwm4 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>If wattless current flows in the AC circuit, then the circuit is :</p> | [{"identifier": "A", "content": "Purely Resistive circuit"}, {"identifier": "B", "content": "Purely Inductive circuit"}, {"identifier": "C", "content": "LCR series circuit"}, {"identifier": "D", "content": "RC series circuit only"}] | ["B"] | null | <p>For wattless current to flow in AC circuit the circuit will be Purely Inductive circuit.</p> | mcq | jee-main-2022-online-25th-june-morning-shift | 8,892 |
1l5bbu66y | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>Given below are two statements :</p>
<p>Statement I : The reactance of an ac circuit is zero. It is possible that the circuit contains a capacitor and an inductor.</p>
<p>Statement II : In ac circuit, the average power delivered by the source never becomes zero.</p>
<p>In the light of the above statements, choose th... | [{"identifier": "A", "content": "Both Statement I and Statement II are true."}, {"identifier": "B", "content": "Both Statement I and Statement II are false."}, {"identifier": "C", "content": "Statement I is true but Statement II is false."}, {"identifier": "D", "content": "Statement I is false but Statement II is true.... | ["C"] | null | <p>$$X = |{X_C} - {X_L}|$$</p>
<p>So, it can be zero if $${X_C} = {X_L}$$</p>
<p>And, average power in ac circuit can be zero.</p> | mcq | jee-main-2022-online-24th-june-evening-shift | 8,893 |
1l5c3toi9 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>A resistance of 40 $$\Omega$$ is connected to a source of alternating current rated 220 V, 50 Hz. Find the time taken by the current to change from its maximum value to the rms value :</p> | [{"identifier": "A", "content": "2.5 ms"}, {"identifier": "B", "content": "1.25 ms"}, {"identifier": "C", "content": "2.5 s"}, {"identifier": "D", "content": "0.25 s"}] | ["A"] | null | <p>$$I = {I_0}\cos (\omega t)$$ say</p>
<p>$$\Rightarrow$$ At maximum $$\omega {t_1} = 0$$ or $${t_1} = 0$$</p>
<p>Then at rms value $$I = {I_0}/\sqrt 2 $$</p>
<p>$$ \Rightarrow \omega {t_2} = \pi /4$$</p>
<p>$$ \Rightarrow \omega ({t_2} - {t_1}) = \pi /4$$</p>
<p>$$\Delta t = {\pi \over {4\omega }} = {{\pi T} \over {... | mcq | jee-main-2022-online-24th-june-morning-shift | 8,894 |
1l5c4iiyq | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>As shown in the figure an inductor of inductance 200 mH is connected to an AC source of emf 220 V and frequency 50 Hz. The instantaneous voltage of the source is 0 V when the peak value of current is $${{\sqrt a } \over \pi }$$ A. The value of $$a$$ is ___________.</p>
<p><img src="data:image/png;base64,UklGRqINAABX... | [] | null | 242 | <p>$${I_{rms}} = {{{V_{rms}}} \over z}$$</p>
<p>$$z = {X_2} = {\omega _2}$$</p>
<p>$$ = 2\pi \times 50 \times {{200} \over {1000}}$$</p>
<p>$$ = 20\,\pi $$</p>
<p>$$\therefore$$ $${I_{rms}} = {{220} \over {20\pi }} = {{11} \over \pi }$$</p>
<p>$$\therefore$$ $${I_{peak}} = \sqrt 2 \times {{11} \over \pi }$$</p>
<p>$$... | integer | jee-main-2022-online-24th-june-morning-shift | 8,895 |
1l5w325pc | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>In series RLC resonator, if the self inductance and capacitance become double, the new resonant frequency (f<sub>2</sub>) and new quality factor (Q<sub>2</sub>) will be :</p>
<p>(f<sub>1</sub> = original resonant frequency, Q<sub>1</sub> = original quality factor)</p> | [{"identifier": "A", "content": "$${f_2} = {{{f_1}} \\over 2}$$ and $${Q_2} = {Q_1}$$"}, {"identifier": "B", "content": "$${f_2} = {f_1}$$ and $${Q_2} = {{{Q_1}} \\over {{Q_2}}}$$"}, {"identifier": "C", "content": "$${f_2} = 2{f_1}$$ and $${Q_2} = {Q_1}$$"}, {"identifier": "D", "content": "$${f_2} = {f_1}$$ and $${Q_2}... | ["A"] | null | <p>We know,</p>
<p>Quality factor (Q factor)</p>
<p>$${Q_1} = {{{w_1}} \over {\Delta w}}$$</p>
<p>$$ = {1 \over {\sqrt {LC} }} \times {L \over R}$$</p>
<p>$$ = {1 \over R}\sqrt {{L \over C}} $$</p>
<p>Now, when $$L' = 2L$$ and $$C' = 2C$$ then $${Q_2} = {1 \over R}\sqrt {{{2L} \over {2C}}} = {1 \over R}\sqrt {{L \over... | mcq | jee-main-2022-online-30th-june-morning-shift | 8,896 |
1l5w3onnm | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>A series LCR circuit with $$R = {{250} \over {11}}\,\Omega $$ and $${X_L} = {{70} \over {11}}\,\Omega $$ is connected across a 220 V, 50 Hz supply. The value of capacitance needed to maximize the average power of the circuit will be _________ $$\mu$$F. (Take : $$\pi = {{22} \over 7}$$)</p> | [] | null | 500 | For maximum power
<br/><br/>$$
\begin{aligned}
&\text { power factor }=\cos \theta=1\\\\
& \therefore \frac{R}{Z}=1 \\\\
&R^{2}=Z^{2} \\\\
&R^{2}=\left(\mathrm{X}_{\mathrm{L}}-\mathrm{X}_{\mathrm{C}}\right)^{2}+\mathrm{R}^{2} \\\\
&\mathrm{X}_{\mathrm{L}}=\mathrm{X}_{\mathrm{C}} \\\\
&\frac{70}{11}=\frac{1}{100 \pi \t... | integer | jee-main-2022-online-30th-june-morning-shift | 8,897 |
1l6dyqydu | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>To increase the resonant frequency in series LCR circuit,</p> | [{"identifier": "A", "content": "source frequency should be increased."}, {"identifier": "B", "content": "another resistance should be added in series with the first resistance."}, {"identifier": "C", "content": " another capacitor should be added in series with the first capacitor."}, {"identifier": "D", "content": " ... | ["C"] | null | <p>Resonant frequency $$ = {1 \over {\sqrt {LC} }} = {\omega _0}$$</p>
<p>$$\Rightarrow$$ If we decrease C, $$\omega$$<sub>0</sub> would increase</p>
<p>$$\Rightarrow$$ Another capacitor should be added in series.</p> | mcq | jee-main-2022-online-25th-july-morning-shift | 8,898 |
1l6f4u1ke | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>When you walk through a metal detector carrying a metal object in your pocket, it raises an alarm. This phenomenon works on :</p> | [{"identifier": "A", "content": "Electromagnetic induction"}, {"identifier": "B", "content": "Resonance in ac circuits"}, {"identifier": "C", "content": "Mutual induction in ac circuits"}, {"identifier": "D", "content": "Interference of electromagnetic waves"}] | ["B"] | null | <p>Metal detector works on the principle of resonance in ac circuits.</p> | mcq | jee-main-2022-online-25th-july-evening-shift | 8,899 |
1l6gmfkts | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>In a series $$L R$$ circuit $$X_{L}=R$$ and power factor of the circuit is $$P_{1}$$. When capacitor with capacitance $$C$$ such that $$X_{L}=X_{C}$$ is put in series, the power factor becomes $$P_{2}$$. The ratio $$\frac{P_{1}}{P_{2}}$$ is:</p> | [{"identifier": "A", "content": "$$\\frac{1}{2}$$"}, {"identifier": "B", "content": "$$\\frac{1}{\\sqrt{2}}$$"}, {"identifier": "C", "content": "$$\\frac{\\sqrt{3}}{\\sqrt{2}}$$"}, {"identifier": "D", "content": "2 : 1"}] | ["B"] | null | <p>$${P_1} = \cos \phi = {1 \over {\sqrt 2 }}({X_L} = R)$$</p>
<p>$${P_2} = \cos \phi ' = 1$$ (will become resonance circuit)</p>
<p>So, $${{{P_1}} \over {{P_2}}} = {1 \over {\sqrt 2 }}$$</p> | mcq | jee-main-2022-online-26th-july-morning-shift | 8,900 |
1l6go8ch7 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>The effective current I in the given circuit at very high frequencies will be ___________ A.</p>
<p><img src="data:image/png;base64,UklGRl4NAABXRUJQVlA4IFINAABwkwCdASoAAwsBP4G61WS2LqunIpHKesAwCWlu/CLd+ANQp2cyuHzB/tfB5h12b/Z3/A/X/6xHm0v7HTa4OeK/+fS6kG1swPbmUmYpzlQe5yqBgz33HKQOvj2RlRQO9ls0lhrP4fhDnhEqmWxUV7mSDLguaaOB3... | [] | null | 44 | <p>Equivalent circuit will be</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l6v6qjj9/0f2354a1-d851-4841-9174-f8c431327613/8f441950-1cd5-11ed-843d-81ad9f680592/file-1l6v6qjja.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l6v6qjj9/0f2354a1-d851-4841-9174-f8c431327613/8f... | integer | jee-main-2022-online-26th-july-morning-shift | 8,901 |
1l6ji7zez | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>A direct current of $$4 \mathrm{~A}$$ and an alternating current of peak value $$4 \mathrm{~A}$$ flow through resistance of $$3\, \Omega$$ and $$2\,\Omega$$ respectively. The ratio of heat produced in the two resistances in same interval of time will be :</p> | [{"identifier": "A", "content": "3 : 2"}, {"identifier": "B", "content": "3 : 1"}, {"identifier": "C", "content": "3 : 4"}, {"identifier": "D", "content": "4 : 3"}] | ["B"] | null | <p>Ratio = $${{i_1^2{R_1}} \over {{{\left( {{{{i_2}} \over {\sqrt 2 }}} \right)}^2}{R_2}}} = {{{4^2} \times 3} \over {{{\left( {{4 \over {\sqrt 2 }}} \right)}^2} \times 2}}$$</p>
<p>$$\Rightarrow$$ Ratio = 3 : 1</p> | mcq | jee-main-2022-online-27th-july-morning-shift | 8,902 |
1l6jj63xi | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>To light, a $$50 \mathrm{~W}, 100 \mathrm{~V}$$ lamp is connected, in series with a capacitor of capacitance $$\frac{50}{\pi \sqrt{x}} \mu F$$, with $$200 \mathrm{~V}, 50 \mathrm{~Hz} \,\mathrm{AC}$$ source. The value of $$x$$ will be ___________.</p> | [] | null | 3 | <p>$${X_C} = {1 \over {wc}} = {{\pi \sqrt x } \over {2\pi \times 50 \times 50}} \times {10^6}$$</p>
<p>$$v_R^2 + v_C^2 = {(200)^2}$$</p>
<p>$$v_C^2 = {200^2} - {100^2}$$</p>
<p>$${v_C} = 100\sqrt 3 \,V$$</p>
<p>$${v_R} = 100\,V$$</p>
<p>$$P = {{{V^2}} \over R}$$</p>
<p>$$R = {{100 \times 100} \over {50}} = 200\,\Omega... | integer | jee-main-2022-online-27th-july-morning-shift | 8,903 |
1l6knirm5 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>A series LCR circuit has $$\mathrm{L}=0.01\, \mathrm{H}, \mathrm{R}=10\, \Omega$$ and $$\mathrm{C}=1 \mu \mathrm{F}$$ and it is connected to ac voltage of amplitude $$\left(\mathrm{V}_{\mathrm{m}}\right) 50 \mathrm{~V}$$. At frequency $$60 \%$$ lower than resonant frequency, the amplitude of current will be approxim... | [{"identifier": "A", "content": "466 mA"}, {"identifier": "B", "content": "312 mA"}, {"identifier": "C", "content": "238 mA"}, {"identifier": "D", "content": "196 mA"}] | ["C"] | null | <p>$$\omega = 0.4{\omega _0}$$ ...... (i)</p>
<p>$$ \Rightarrow I = {V \over Z} = {{50} \over {\sqrt {{R^2} + {{\left( {\omega L - {1 \over {\omega C}}} \right)}^2}} }}$$ ..... (ii)</p>
<p>$$ \Rightarrow I = 238$$ mA</p> | mcq | jee-main-2022-online-27th-july-evening-shift | 8,904 |
1l6mao9vk | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>The equation of current in a purely inductive circuit is $$5 \sin \left(49\, \pi t-30^{\circ}\right)$$. If the inductance is $$30 \,\mathrm{mH}$$ then the equation for the voltage across the inductor, will be :</p>
<p>$$\left\{\right.$$ Let $$\left.\pi=\frac{22}{7}\right\}$$
</p> | [{"identifier": "A", "content": "$$1.47 \\sin \\left(49 \\pi t-30^{\\circ}\\right)$$"}, {"identifier": "B", "content": "$$1.47 \\sin \\left(49 \\pi t+60^{\\circ}\\right)$$"}, {"identifier": "C", "content": "$$23.1 \\sin \\left(49 \\pi t-30^{\\circ}\\right)$$"}, {"identifier": "D", "content": "$$23.1 \\sin \\left(49 \\p... | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l6z6wn42/89ab94f0-99d2-49fe-acfa-1f6b868af330/5b7bf350-1f09-11ed-8ba4-19601952a51f/file-1l6z6wn45.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l6z6wn42/89ab94f0-99d2-49fe-acfa-1f6b868af330/5b7bf350-1f09-11ed-8ba4-19601952a51f... | mcq | jee-main-2022-online-28th-july-morning-shift | 8,905 |
1l6mbcsqo | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>The frequencies at which the current amplitude in an LCR series circuit becomes $$\frac{1}{\sqrt{2}}$$ times its maximum value, are $$212\,\mathrm{rad} \,\mathrm{s}^{-1}$$ and $$232 \,\mathrm{rad} \,\mathrm{s}^{-1}$$. The value of resistance in the circuit is $$R=5 \,\Omega$$. The self inductance in the circuit is _... | [] | null | 250 | <p>$${i \over {{i_{\max }}}} = {1 \over {\sqrt 2 }}$$</p>
<p>$$ = {{{{{V_0}} \over Z}} \over {{{{V_0}} \over R}}}$$</p>
<p>$$ \Rightarrow {R \over Z} = {1 \over {\sqrt 2 }}$$</p>
<p>and $${1 \over {212C}} - 212L = 232L - {1 \over {232C}}$$</p>
<p>so $$212L = {1 \over {232C}}$$</p>
<p>so $${R \over {\sqrt {{R^2} + {{\le... | integer | jee-main-2022-online-28th-july-morning-shift | 8,906 |
1l6p51yr7 | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>An alternating emf $$\mathrm{E}=440 \sin 100 \pi \mathrm{t}$$ is applied to a circuit containing an inductance of $$\frac{\sqrt{2}}{\pi} \mathrm{H}$$. If an a.c. ammeter is connected in the circuit, its reading will be :</p> | [{"identifier": "A", "content": "4.4 A"}, {"identifier": "B", "content": "1.55 A"}, {"identifier": "C", "content": "2.2 A"}, {"identifier": "D", "content": "3.11 A"}] | ["C"] | null | <p>$$I = {V \over {\omega L}}$$</p>
<p>$$ = {{440} \over {100\pi \times {{\sqrt 2 } \over \pi }}} = {{44} \over {10\sqrt 2 }}$$</p>
<p>$$ \Rightarrow {I_{rms}} = {I \over {\sqrt 2 }} = {{44} \over {20}} = 2.2\,A$$</p> | mcq | jee-main-2022-online-29th-july-morning-shift | 8,907 |
ldo77jpr | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | An alternating voltage source $\mathrm{V}=260 \sin (628 \mathrm{t}$ ) is connected across a pure inductor of $5 \mathrm{mH}$ Inductive reactance in the circuit is : | [{"identifier": "A", "content": "$6.28 \\Omega$"}, {"identifier": "B", "content": "$0.318 \\Omega$"}, {"identifier": "C", "content": "$0.5 \\Omega$"}, {"identifier": "D", "content": "$3.14 \\Omega$"}] | ["D"] | null | $\omega$ = 628 rad/s
<br/><br/>$X_{L}=L \omega$
<br/><br/>$$
\begin{aligned}
& =5 \mathrm{mH} \times 628 \\\\
& =3.14 \Omega
\end{aligned}
$$ | mcq | jee-main-2023-online-31st-january-evening-shift | 8,909 |
ldo7gfpq | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | A series $\mathrm{LCR}$ circuit consists of $\mathrm{R}=80 \Omega, \mathrm{X}_{\mathrm{L}}=100 \Omega$, and $\mathrm{X}_{\mathrm{C}}=40 \Omega$. The input <br/><br/>voltage is 2500 $\cos (100 \pi \mathrm{t}) \mathrm{V}$. The amplitude of current, in the circuit, is _________ A. | [] | null | 25 | $\omega=100 \pi$
<br/><br/>$$
\begin{aligned}
& \text { So } Z=\sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}} \\\\
& =\sqrt{80^{2}+(100-40)^{2}} \\\\
& =100 \Omega \\\\
& i_{0}=\frac{V_{0}}{Z}=\frac{2500}{100} \mathrm{~A}=25 \mathrm{~A}
\end{aligned}
$$ | integer | jee-main-2023-online-31st-january-evening-shift | 8,910 |
1ldohdscu | physics | alternating-current | ac-circuits-and-power-in-ac-circuits | <p>A series LCR circuit is connected to an ac source of $$220 \mathrm{~V}, 50 \mathrm{~Hz}$$. The circuit contain a resistance $$\mathrm{R}=100 ~\Omega$$ and an inductor of inductive reactance $$\mathrm{X}_{\mathrm{L}}=79.6 ~\Omega$$. The capacitance of the capacitor needed to maximize the average rate at which energy ... | [] | null | 40 | To maximize the average rate at which energy
supplied i.e. power will be maximum.
<br/><br/>So in LCR circuit power will be maximum at the
condition of resonance and in resonance condition
<br/><br/>$$
\begin{aligned}
& \therefore X_{L}=X_{C} \\\\
& 79.6=\frac{1}{2 \pi(50) \times C} \\\\
& C=\frac{1}{79.6 \times 2 \p... | integer | jee-main-2023-online-1st-february-morning-shift | 8,911 |
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